Generally they were pretty effective which is why wooden earthworks and palisades have been used throughout history, from the Greeks and Romans to at least the 19th century and later. These have been structured in many different ways to serve many different purposes, from Julius Caesar's system of earthen siegeworks during his conquest of Gaul, to early Norman motte-and-bailey castles, to gunpowder armies. There's nothing completely typical about how these were constructed or what an infantry assault on them was like but I can cite a few examples.

First is Julius Caesar's description of the earthworks he built in 52 BC while besieging Alesia taken from his book The Gallic War. [EDIT: This was actually the Siege of Avaricum also in 52 BC and the following description of earthworks weren't entirely made of wood as you can see since they included stones, but there were ones entirely made of wood during these same operations]

At right angles to the wall, along its entire length, beams are placed in the ground at equal intervals, two feet apart. These are secured firmly from the inside, and then covered with large amounts of ruble; the gaps we mentioned between are filled with large rocks at the front. Once the beams are laid and joined, another level is added on top; the length of the gaps remains the same, but the second set of beams does not touch the first. Rather, because the spaces in between the means are the same, each one is firmly kept separate by single stones. The whole construction is built in this manner until the walls reach the appropriate height. The finished edifice is not unattractive, with the variations in its appearance and the alternating beams and rocks in straight lines following the line of their own level. Moreover, it offers an excellent means for practical defense of cities. The stones give protection from fire and the timber from battering rams--for it is impossible to break through continuous beams, usually forty feet long and secured on the inside, or to tear them apart.

Caesar goes on to describe how the Gauls would frequently make "sorties by day and night, trying to set fire to the earthwork or attack our soldiers while they were working on the siege" and they even tunneled underneath the palisades in order to set them on fire from below. However, ultimately none of these attempts were successful and the starving Gauls were eventually forced to surrender to the Romans.

Another description of an attack against wooden palisades I know of occurred in 1671 when a force of 400-600 English and French buccaneers attacked a fortress in Panama held by 314 Spanish soldiers and allied Indians. This account is from a book called The Buccaneers of America published in 1678 by Alexandre Exquemelin who was an eyewitness to these events. They ended up taking the fortress but with very heavy casualties.

Three days after leaving St. Catalina the four ships sighted the fortress of Chagre, standing on a high mountain at the mouth of the river. The summit is intersected by a moat, a good thirty feet deep, which can be crossed at one place only, where there is a drawbridge. The fortress is surrounded by strong palisades filled in with earth. Four bastions command the landward side, and two face the sea. ...

...

When they did reach the fortress, they were instantly greeted by intense artillery fire from the Spaniards within. The rovers suffered heavy damage, as they were completely exposed, for in order to storm the fortress they had to break cover. The defenders could see them from head to foot, whereas the rovers were unable to get a glimpse of the Spaniards. They were in great anxiety, not knowing how best to set about the attack. This was going to be a hard nut to crack, yet they dared not turn back and be disgraced in the eyes of their fellows. They resolved to assault the fortress, come what may.

They made a vigorous assault with muskets and hand grenades, but the Spaniards had such good cover the rovers could do them little damage. They returned the fire furiously with cannon and muskets, shouting at the attackers: 'Let the others come too, you English dogs, enemies of God and the King -- you shall not get to Panama!' Finally, the buccaneers were forced to retire.

In the evening they advanced again, intending to throw in hand-grenades and then scale the palisades, but failed in the attempt. One of the buccaneers was pierced through the shoulder by an arrow [fired by an Indian allied with the Spanish]; in a fury, he wrenched it out, took a wad of cotton from his pouch, tied it to the arrowhead and set fire to it. When it was well alight, he stuck the arrow in his musket and shot it into the palm-leaf thatch of some houses within the fortress walls. The other buccaneers, seeing this idea, began to do the same. At last, they succeeded in setting the roofs of two or three houses on fire.

The defenders apparently were so busy they did not notice this until the burning thatch came crashing down on them, igniting a cask of gunpowder and putting most of the Spaniards out of action. The buccaneers made the most of their opportunity and renewed the assault. The Spaniards were doing all they could to extinguish the fire, but could not get sufficient water to check the spreading blaze, for the buildings were nearly all of dry wood. A fresh breeze was blowing, which did not help matters.

Meanwhile the buccaneers, seeing the fire increase within the fortress walls, endeavored to set the outer palisades ablaze. The attempt succeeded, although not without great trouble and loss of many men, for when the Spaniards realized their enemies were in the moat, they hurled down pots of gunpowder with lighted fuses attached. These missiles worked great havoc among the attackers, but they achieved their objective notwithstanding. By nightfall the palisades were burning. The rovers crept on hands and knees close to the blaze, shooting down every Spaniard they glimpsed through the flames.

At daybreak the fire had burnt through the palisades almost completely. The earth rampart which they had supported started to collapse, and as the ground gave way under the guns they too pitched down into the moat. The buccaneers could concentrate their fire on the Spaniards now exposed to view. The governor of the fortress kept his men under such discipline that they dare not withdraw; he had artillery brought into the breach made by the fire so that resistance could continue. Yet the defenders' spirits sank, now they no longer had proper cover. The rovers kept up such a furious accurate fire that any Spaniard who dared show himself was instantly shot down.

All this time the fire had been spreading. Once the breach was wide enough, the rovers too tried to quell the flames, fetching earth for the purpose. While one group was busy at this work, the rest attended to the Spaniards. Finally, most of the defenders succumbed, either from the fire or from the rovers' deadly bullets. About midday the buccaneers sprang through the breach, despite the fire, which was still raging, and despite the continued resistance of the governor and some twenty-five of his men. Those whose guns were now useless hurled stones or fought with spears, yet the buccaneers pressed on notwithstanding and at last made themselves masters of the fortress.

At this point Exquemelin claims most of the surviving Spaniards refused to surrender and fought to the death except for a few who they captured to get information from. He says 100 buccaneers were killed and 70 wounded in the assault, while the Spaniards had only 30 survivors, 20 of whom were wounded. Another eight or nine are said to have escaped.

They could be extremely effective, particularly when placed on well-constructed earthworks. In New Zealand, Māori pa were constructed with palisades and embankments, but I'll just talk about a single event: the British attack on Ōhaeawai pa in 1845.

Very briefly, the Northern War began when Hone (John) Heke, a Māori chief that had become dissatisfied with the expansion of British sovereignty and influence, and the failure to uphold the promises of the treaty of Waitangi, cut down the British flag at Kororāreka three times, on the third occasion defeating a garrison that had been installed there and burning the town to the ground (Kororāreka was not particularly big, but the bulk of the settlement of New Zealand hadn't happened yet, so it was still one of the largest European towns in New Zealand, and was also an important stopover for the American whaling industry). In June 1845 the British force sent to defeat Heke, and their Māori allies under Tāmati Waka (Thomas Walker) Nene attacked the pa (fort) of Hone Heke's uncle, and main supporter, Te Ruki Kawiti (very much no Christian name).

Ōhaeawai was extremely strongly defended, the culmination of a half-century worth of influence from firearms on traditional designs, coupled with various innovations of Kawiti's to cope with the artillery they would be facing. You can see a picture here. The pa was oblong, a bit over 100 meters long, with salients providing fields of fire along the front of the walls. The walls were made of a double line of palisades. The first line, the pekerangi (which in later pa evolved into a very light fence designed to slow attackers within short range of the main defence) was a pretty substantial obstacle in its own right at Ōhaeawai. It was almost the same height as the interior palisade, made of small tree-trunks sunk into the ground at regular intervals with the gaps between them filled with a fence of lighter branches lashed together with vines. The whole thing was then covered in a thick flax matting that was capable of deadening bullets, and even cannonballs to some extent. The matting was also very flexible, and would close up behind the cannonballs, making it very difficult to see how much damage was being done. About three feet inside this fence was the main palisade, which was made almost entirely of reasonably large puriri trunks sunk deeply into the ground (puriri is a hard wood that doesn't splinter much). After the battle two six pound cannonballs were found imbedded in the palisade. Behind this was a trench, and the main palisade was loopholed at ground level, so that the defenders, firing out, would be exposed as little as possible. Here's a picture of what the whole thing looked like from the outside. To prevent flanking fire, the main trench was interrupted every few meters by a series of traverses (pakiaka), so the defenders could make their way entirely around the pa zigzagging along the trench, but it couldn't be fired along. Fire has usually been one of the main weaknesses of a stockade, like in the example u/Elphinstone1842 gives above, and was one of the very few ways that pre-European pa were often taken, but the huts at Ōhaeawai were not only fire-proof, but essentially bomb-proof. They were partially sunk in the ground, and their roofs were covered in earth and bundles of green flax. In addition there were deep pits entirely covered in timber and earth.

After a weeks bombardment with his four 6 and 12 pound guns, the 32 pound ship's gun that Colonel Despard had requested arrived, but a sally that morning by the defenders convinced him to try an assault. 220 soldiers and sailors formed an assault party (there were probably no more than a hundred defenders in the pa at this point). Despard would later claim that the pioneers had left behind the axes and ropes that they'd been issued, though he later withdrew this claim. It certainly seems that they only arrived with a single ladder though. When the attackers were within fifty paces of the pa the Māori began firing. Upon arriving at the palisade it turned out that it was nowhere near as damaged as had been expected. The pioneers with axes and ropes, and the sailors with their cutlasses attempted to bring down the wall, but any damage they did to the pekerangi simply left them more exposed to fire from behind the main palisade that they were unable to bring down. The single ladder was set up, but it would have been suicide to try climbing it. After about five minutes of fruitless work, exposed to point blank fire from in front of them, and enfilading fire from the salient to their left, the storming party were called back, leaving nearly forty dead and taking back seventy wounded.

what would it take to break the defenses

A lot. About a week later Kawiti and his men abandoned the pa, which was their usual custom after blood had been spilt in a pa, and, after they realised, the British and their Māori allies took possession (which allowed Despard to claim victory, though this was not a perception shared by many others) and destroyed the pa. This work took three days, several large fires were built against the walls, and some of the posts that were too large to be burnt while upright were pulled down "In some cases it took the whole strength of forty men, with ropes, to pull down one post, although much of the earth at the base had been previously dug away for the purpose of loosening it". It was fairly clear that, whatever the truth of Colonel Despard's claims about the failures of the assault party were, it didn't really matter: all that more ladders and axes could have done would have been to multiply the British casualties, anyone who had made it inside the pa would have been shot from cover from the many bunkers and disconnected sections of trench.

TLDR No: no you can't take an unbreached pa with infantry. Even artillery weren't as useful as you'd expect.

For further reading there are several primary descriptions of Ōhaeawai pa here in 'the Pa Maori' by Elsdon Best.

There's a description of the assault here, in 'The New Zealand Wars' by James Cowan. It's quite old, and in a highly dramatised style, but there are quotes from veterans of the action (on both sides) that make it worth reading.

Sources: the two I mentioned, but mainly 'The New Zealand Wars' by James Belich, and 'Wars Without End' by Danny Keenan.