The Templars and Hospitallers continued to fight the Turks and Saracens after 1291, as the crusades didn't end with the loss of Acre that year, but rather they continued throughout the rest of the medieval period and even beyond. With the final loss of the Levant in 1291, the military orders of the Templars, Hospitallers, Lazarites, and Order of St Thomas all retreated to Cyprus (the Teutonic Order also went there, but moved its headquarters to Venice and had already shifted its main focus of operations to the Baltic). The last two of these soon demilitarised but the Hospitallers and Templars continued to maintain a military role against the Muslim polities of Anatolia and the Levant.

The Templars' first (and only) real military achievement after 1291 was the capture of Ruad in 1300. The Templars, alongside the Hospitallers, Teutonic Knights, and the Kingdom of Cyprus, attempted to retake Tortosa on the mainland. They raided the city but failed to hold it and instead captured the tiny island of Ruad off the coast, which the Templars held until it was lost to the Saracens in 1302. The Order still planned to continue its military campaigns though. The master of the Temple and of the Hospital both visited Cilician Armenia in 1293/94 to rally support and in 1306 Jacques de Molay, the last Templar grandmaster, produced a crusade plan at the pope's request. However, these plans were cut short by the Order's arrest from 1307 and its suppression in 1312.

The Hospitallers soon moved their headquarters from Cyprus, conquering Rhodes from the Byzantines in 1306-10. From that island they conducted a largely naval war against the Turks until they were expelled from the island in 1522. They made some incursions into the mainland as well, holding Corinth in the 1390s and Smyrna in Turkey from 1374-1402 and a castle at Bodrum from 1402-1523. After Rhodes, the Hospital settled on Malta in 1530, from which they again conducted a naval war against Muslim shipping and defeating a Turkish siege of the island in 1565. They made brief forays into Africa as well. The Order was given Tripoli in Libya in 1530, which they lost to the Turks in 1551. The Hospitallers finally demilitarised after 1798, when they were expelled from Malta by Napoleon.

It should be noted that these orders also fought against fellow Christians as well. In Europe, the local heads of military orders were often employed by the relevant king as diplomats, administrators, and commanders. The Hospitallers and Templars served in Wales, Scotland, and Ireland in English armies from the late thirteenth century. The Hospitaller Prior of France fought and died against the English at Crecy. However, both orders continued their wars against Muslim states first and foremost until they were suppressed (the Templars) or demilitarised (the Hospitallers).

I have written a bit more about the late medieval and modern history of the military orders here

Sources:

Anthony Luttrell, 'The Military Orders, 1312-1798', in Jonathan Riley-Smith (ed.), The Oxford Illustrated History of the Crusades (Oxford, 1995), pp. 326-64.

Helen Nicholson, The Knights Hospitaller (Woodbridge, 2001).

Desmond Seward, The Monks of War (London, 1972).