The military orders certainly did fight and compete with each other. Ideally, the Orders were supposed to work together as they served the same master and had the same goals. The various military orders of the Crusades were all subordinate to the Pope and fighting to preserve the Crusader States. In practice however, the orders became powerful landlords and vassals, leading to them being entangled in politics and personal quarrels sometimes bringing them into conflict with each other.
Sometimes legal disputes could cause conflict between the orders. There was a long-running one between the Hospitallers and Teutonic Knights. The Hospitallers mistakenly believed that the Teutonic Order was based off a German hospital in Jerusalem, one which the Hospitallers were entitled to control. The Teutonic Knights were actually founded as a German hospital in Acre. Because of this error, the Hospitallers said that they should be placed in charge of the Teutonic Order. Beginning in 1190, this dispute did not end until 1231, when the Hospitallers asked the Teutonic Knights to act as negotiators for them in another legal case in Italy. However, after this date relations were generally cordial. The two orders acted on each others' behalf and guarded each others' holdings. They also fought on the same side against the Templars in the civil war of the 1240s. These property disputes could lead to open conflict, such as in 1197 when the Templars and Hospitallers fought over their rights to an estate in Tripoli.
In rarer cases, the military orders' secondary role as secular landlords could draw them into civil wars. In 1228 the Holy Roman Emperor Frederick II arrived in the Holy Land. He claimed to be the king of the Kingdom of Jerusalem by the right of his wife, Isabella II of Jerusalem, who died that year. However, the nobility of the Kingdom had instead proclaimed that Frederick and Isabella's young son, Conrad II, would be king. After a short campaign and a series of negotiations, Frederick secured the handover of Jerusalem to Christian control but also antagonised the nobles of the Kingdom, provoking a civil war. The nobles recognised Frederick's rights as regent for his son, but not his right to be king himself. Frederick soon returned home, leaving some Italian supporters to represent his interests in the East.
Intermittent conflicts bubbled away over the following years. In 1241 the Templars attacked the properties of the Hospitallers and Teutonic Knights in Acre because these orders had supported a treaty with Egypt, one that the Templars rejected. War broke out in 1242. Conrad II, then living in Europe, wrote to the people of his kingdom and said that he was now of age and so he had appointed an Italian, Richard Filangieri, to be his regent. This angered the nobility as they believed that the regent needed to be a relative of Conrad's from within the Crusader States, Alice of Champagne. The powerful Ibelin family backed her claims. In addition to this, the Ibelins and Templars supported making peace with Damascus and going to war with Egypt, whilst Frederick's imperial supporters and the Hospitallers wanted peace with Egypt and war with Damascus. With the Templars, church, and nobles of the Crusader States led by the Ibelins fighting for Conrad, and Filangieri, Hospitallers and Teutonic Knights fighting for Frederick.
Filangieri arrived in the Holy Land in early 1242 and stayed with the Hospitallers in Acre. The Order's stronghold was soon besieged by Ibelin, the Templars, and his supporters, forcing Filangieri to flee abroad. The Ibelins continued to fight the imperial faction, driving them out from their last stronghold of Tyre in 1243. The Crusader States then went to war with Egypt, were defeated, and lost control of Jerusalem.
Another war broke out between the Temple and Hospital in 1256, ending in 1258. This was triggered by a property dispute between the Genoese and Venetian communes in the Holy Land. The Hospitallers backed Genoa, the Templars Venice. Venice won, though it there's no evidence to show that the two orders directly fought each other this time.
The military orders sometimes even fought within themselves. One of the early modern grandmasters of the Hospitallers tried to enact a series of moral reforms, expelling prostitutes and the brethren's girlfriends into the countryside of Malta. In response, the brethren overthrew the grandmaster.
The smaller military orders like the Order of St Lazarus (the Leper Knights) and the English Knights of St Thomas of Acre don't seem to have been involved in these conflicts as heavily, probably because they were minor lords compared to the Temple, Hospital, and Teutonic Knights. In addition to fighting each other in the East, in Europe the military orders often fought other Christians. The orders constantly had to balance their role as ecclesiastics and as secular landlords, often failing to avoid civil conflicts. John Langstrother, the Hospitaller Prior of England, was executed in 1471 in the Wars of the Roses for backing Henry VI against Edward IV. The Scottish master of the Temple fought and died for Edward I against William Wallace at Falkirk in 1298.
Sources:
Helen Nicholson, The Knights Hospitaller (Woodbridge, 2001).
Nicholas Morton, The Teutonic Knights in the Holy Land, 1190-1291 (Woodbridge, 2009).
Desmond Seward, The Monks of War (London, 1972).
The Crusaders, despite often being depicted as this sort of pan-European initiative, were frequently fractious and prone to prioritizing petty politics over working together. The knightly orders were no different.
One particular instance of this was the Siege of Ascalon in 1153. Ascalon, which is perhaps best described as the Egyptian counterpart to Krak des Chevaliers, proved a frequent thorn in the side of the Crusaders. And just as the "Saracens" would eventually decide to strike at Krak des Chevaliers, the Crusaders eventually decided they were tired of the incessant raids from Ascalon and they would just take the damned thing and be done with it.
Which, of course, is easier said than done. As described by Sir Steven Runciman, "Ascalon was a tremendous fortress, spreading from the sea in a great semicircle." This was a problem as the Crusaders didn't have naval superiority. The Egyptians were able to sail a fleet into the fortress in June (the siege having begun in February) without the Christian fleet even attempting to stop them. The inability to close off the city meant that it was going to be nigh impossible to starve it out. The risk of a relief force arriving meant that a multiple year siege was a non-option. And that left the Crusaders with only one choice: take the city by storm.
Ascalon, however, had some pretty legitimate fortifications. Perhaps not quite as impressive as Tyre, where troops from Damascus were able to force the Crusaders to retreat before they reduced Tyre's defenses to a state that made it vulnerable to escalade, but they were well-maintained and would take several months of concerted effort to break.
At this point in the siege a particularly bizarre sequence of events unfold: the Crusaders had invested most of their resources in an enormous siege tower (perhaps remembering how successful a pair of huge towers proved to be at Tyre?). The defenders managed to conduct a successful sortie and burn the thing down.
At some point in the night, presumably after the foundations had been weakened by the firing, wind blew the tower over, whereupon it crashed into Ascalon's walls. From there, the intense heat of the fire cracked and undermined the walls and ramparts. By the time the next day rolled around, the wall had a breach. The Templars, perhaps believing that this was God's will (how else would they have explained a siege engine, lost on their stretch of wall, opening a breach?), decided that the honor of the attack should go them and them alone. They attacked and they failed spectacularly.
At this point the Christian forces were largely demoralized. Their great siege engine was burned, the Templars had lost dozens of their finest knights, and the breach was, apparently, unassailable. As the Crusader army wavered, it was the Hospitallers who argued that another effort should be made. This instance of the Hospitallers agreeing with the Templars was apparently so surprising that it swayed the barons and lords that were on the fence, another effort was made, and Ascalon fell. The treatment of the historical records of this event suggests that it was extremely unusual for the rival orders to agree on much of anything, which made the Hospitaller argument all the more poignant.
The Teutonic Knights are a bit more... odd. They didn't come into existence until around the time of the above siege (their roots start about a decade prior). For their first fifty years or so, they were simply a branch of the Hospitallers. The Hospitallers and Templars tended to boast an unusually high percentage of "Franks," so Pope Celestine II decided there should be a German organization of some sort, which he originally flagged as a priory under the Hospitallers. Over time, this eventually evolved into a full-blown military order. But the Teutonic Knights showed up relatively late; their first holding of note wasn't until 1229, shortly after the recapture of Jerusalem, that they'd have a castle worthy of the term. But Jerusalem fell (again) in the 1240's, with this Teutonic castle meeting a similar fate a few decades later. And unlike the 1187 siege by Saladin, there would be no subsequent crusader relief (not that the complete disaster that was the Fourth Crusade helped). The Teutonic Knights are best known for their role in the Northern Crusades and subsequent wars with (then-Christian) Lithuania, not their relatively brief stint in Outremer.
As for actual battles between the Templars and Hospitallers? I'm not aware of any. They certainly spent a great deal of time and energy trying to outmaneuver or one-up one another (I suspect that trying to out-do the Templars was a contributing factor in the enormous expenses incurred in building Krak des Chevaliers), but I don't believe they ever fought one another in earnest. There were always Muslim armies to fight; fighting fellow Christians was a luxury the perpetually outnumbered knights of Outremer didn't have.
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I ends up asking my professor about this one. In general, it was "god's will" for the knights, so fighting amongst eachother wasn't really a thing. That's also why it was rare for lords to take another absent lord's land while he was gone in the crusades. Some conflict can be seen between the knights over some morally questionable acts, such as the siege and taking of Constantinople.