I talk a bit about Japanese castles throughout time here, with a bit at the end on Sengoku-era castles. Leading on from that, a notable factor of Sengoku warfare was the distinct lack of a special field of siege weapons - mangonels/catapults were used, however more during the battlefield rather than at castles, and cannons were few and far between due to their relative expense - a big reason for why these weapons didn't exist is implicitly answered in the link above: the fact that, for the most part, they relied on natural fortifications of mountains and built-in mazes, rather than a massive stone wall. And for this reason, the siege itself focused more on the siege part than the warfare part (on that note, the word siege looks funny now) - essentially, maintaining proper supply lines into enemy territory, the morale of the troops (who at this time comprised largely of ashigaru, and so the seasonal differences need to be accounted for), and the actual cutting off from reinforcements and supplies for the defenders. The actual warfare itself often involved the two sides meeting in a peaceful situation and discussing their terms, as well as who had the better chance of winning should it come to an actual assault - forget the whole thing about bushido and honour, if, as the commander of a castle you could ensure all your men survived by surrendering and joining the other side, then by all means you would do so. By the same token, if you figured you had a fighting chance in defending, then you would also defend. So, for the most part, siege warfare in Japan looked similar to that elsewhere in terms of the factors that defined how long the siege went for and the motives of both sides - that due in part to the purpose of the siege and the castle/fortification itself. I should however qualify that statement to say that, for the most part, sieges were avoided due to the fact that they were unproductive for both sides - neither did anything and forced their soldiers to commit, which was not as preferred as a decisive engagement (either through marching out on to the battlefield to fight, or meating peacefully as mentioned above). So TLDR it would have looked similar to siege warfare in Europe minus fancy technologies, the actual assault would be painful, and because of that most of the time sieges were avoided when possible.

So I am actually REALLY PISSED at the quality of answer I am about to write. I don't like relying only on Turnbull and Osprey (anyone who read my stuff before knows that) so I actually bought two Japanese books off Kindle just for this thread and I am sorry to say that I can't do better. Most of the books are on the castle construction and defences themselves. Even after that, the overall war strategic situation the that the siege is a larger consideration that the tactics of the siege itself. I wish I had a Japanese University paywall login or live in Japan so I wouldn't be limited to Kindle. Also /u/bigbluepanda is correct so there will be repeats. With that off my chest, here goes:

Japanese castles are divided roughly into two kinds. The yamajiro (mountain castle) is built into a mountain. The main defence is actually the mountain itself. Often walls and paths are partly cut out of the mountain itself. The shape of the castle is often rings from one top to another (Turnbull call this “tiered”). The hirajiro (plains castle) is built on a plain. As such it relies much more havily on artificial defences like big walls, large, winding, often watered moats and often incorporate river into the moat system. It uses concentric rings. A third type, the hirayamajiro (“flat-mountain” hill castle) is basically a borderline between the two (the distinction was used in the Edo, but it’s a bit unclear to modern scholars where to draw the line).

A large army could actually quickly overwhelm a castle. Shingen famously took thirty six castles in one day. Even allowing for exaggerations and the fact that a lot of these must have been tiny garrisons losing hope and quickly surrendering, it’s still impressive.

If the castle did not immediately surrender or fall, we could set up for a siege. The castle would be surrounded and cut off from supply. However, if it doesn’t work and the castle is well defended, often time is not actually on the besieger’s side. The feudal levy army of the Sengoku can only stay in the field so long and must get back to their fields or other work, and a large army staying in one place takes up a lot of supplies. So often the besieger is actually forced to assault, or just give up, or assault and then give up. Hojo Ujiyasu famously held off both Takeda Shingen and Uesugi Kenshin (separately of course) in his Odawara Castle, but Kenshin tried only for a dozen or so days and Shingen tried for only four before giving up. Shingen also gave up his assault on Takatenjin Castle.

Once we get to the assault, the troops will probably be scaling walls. Neither the attacker nor the defender is likely to fight over the entire length of the walls at once, but to concentrate on a single section. A lot of yamajiro are designed so that the attacker can physically only attack from one direction anyway. Large complicated siege engines seem to be fairly rare and not nearly the sophistication of mainland Eurasia, though archery towers and simple rams existed and large shields often on wheels to aide approach was common. We are not actually sure how they scaled walls, but ladders are not mentioned often. We don’t know if that’s because ladders are too common writers assume readers know they were used, or that soldiers were expected to scale the walls barehand, or just that scalable walls (which rules out a lot of yamajiro) are actually rare. Luis Frois apparently reported during a siege in 1584 the besieger used a bunch of short sharp sticks, stuck them into the rock wall, and used that as ladder. For what it’s worth, Tokugawa Ieyasu ordered a bunch of ladder for the Siege of Osaka. And scaling likely happened because the castle walls are designed to have difficult footing for scaling.

So whether or not the walls were scaled, fire arrows would be shot and flaming pots thrown. This is probably actually effective since a lot of the architecture is wood. Defenders would throw down rocks and stuff, shoot arrows and fire arquebuses. Should the attacker get over the wall somewhere to open the gate, or the gatehouse is broken in (side note, it seems to be Japanese castles are a lot easier to physically break into, maybe because of the wood construction, but we’re not done yet), Japanese castles are designed like a winding narrow road. Twists and turns breaks up the momentum and make them take a long time to cover a short distance. The paths are also designed so attackers would be subject to fire, often concealed, from multiple directions. The attackers, crowding, lacking moment, being caught in a crossfire, often breaks, allowing the defenders to pursue. In this way, a Japanese castle need to be taken section by section often at high cost. This section on the actual assault is probably the only real difference between Japan and Europe. But the end result is the same: the cost of taking a well defended castle is huge and very time consuming. A lot of these sieges we see garrisons holding out (until attacker giving up, relieved, or surrender) against a force ten times the size.

There are situational methods of avoiding this. Bribing a defender is always good. Tunneling could be used to mess up with the castle’s water supply (apparently not to enter the castle itself), which the Takeda often employed their gold miners to do on campaign. A yamajiro’s well often need to be dug far deeper, allowing the besieger’s diggers at lower elevation to cut the well off. Toyotomi Hideyoshi and Ishida Mitsunari also famously flooded plain castles built in swamp areas for defense (Takamatsu and Oshi respectively, the former succeeded the latter failed). The (successful) flooding cut off any supply and relieve of the castle, destroyed supplies and morale. On an unsourced personal hypothesis, I think it might have also greatly evened the playfield because no more walls to climb.

Negotiations will continue throughout the assault. After close to a month of continuous assault by Takeda Katsuyori, the defenders of Takatenjin Castle had lost maybe 3/4 (me judging by the map) of the castle and decided to surrender due to relief not coming. Takamatsu surrendered before Hideyoshi launched a final assault (he actually needed to scram because Nobunaga got killed, but bluffed his way to the castle’s surrender) and before Mori’s relief arrived. Whether or not relief was coming was extremely important in negotiation, and the arrival of the relief force caused many a decisive battle (Oriyama, Kawagoe, Toishi, Okehazama, Nagashino, etc). Besieger sometimes tried to force a battle by trying to lure a relieve force. After Nagashino, Tokugawa Ieyasu besieged Takatenjin. Both sides built supporting forts (see below) but the end result was that supplies couldn’t get to Takatenjin, and after months of siege Takatenjin offered to surrender. The surrender was refused. Takatenjin would fall soon anyway, and an opportunity was seen to use the siege to lure Katsuyori out to be destroyed once and for all. But Katsuyori could not take gather a relief army anyway and didn’t come in force. In the end the garrison sallied out and was slaughtered.

A lot of time a much longer investment is needed to starve out the castle, but the besieging army can not stay there (see above). In that case forts/redoubts/castles would be constructed to act as area control to keep the encirclement or tighten it without the need of a large force. The defender would try to secure their own supply lines by building supporting forts and if possible also use these to harass the besieger’s lines. The besieger would tighten their grip or try to take these places and turn them against the besieged, and vice versa. A ring of forts tight enough could eventually starve out a castle, and could be used as a platform for the eventual assault. In this way Nobunaga was able to take Odani Castle after three years (eventually by assault) and Hideyoshi was able to take Miki Castle after two and a half years (by starvation). Hideyoshi’s Odawara campaign militarily consisted of mainly surrounding Odawara and then taking its supporting castles one by one.

Finally I need to mention the special case of the winter siege of Osaka, when the castle was intimidated into negotiation/surrendering by pounding from imported cannons.

Sorry I couldn’t give more information on the assault phase itself. I couldn’t find any more.

Sources: Japanese Castles 1540–1640 – Stephen Turnbull ^his ^non-politics, ^non-military-campaign ^stuff ^are ^good ^enough ^to ^use ^okay? 2003
信長公記 (The chronicle of Lord Nobunaga) – Ota Gyuichi (1527—1610?), trans. and edited by J.S.A. Elisonas and J.P. Lamers (2011)
戦国時代の計略大全 (List of Tactics/Strategies of the Sengoku Period) – 鈴木眞哉 (Suzuki Masaya) 2011
城取りの軍事学 (Military Studies of Taking a Castle) – 西股総生 (Nishimata Fusao) 2013
戦国の城 (Castles of the Sengoku) – 小和田哲男 (Owada Tetsuo) 2011

EDIT: See here in regards to the castle town. TL;DR, if it's unprotected the besiegers burns the place down.