Yes, this happened frequently during the First Crusade, mostly with towns and cities that had a majority Christian population already, but sometimes with Muslim-majority cities as well.

The settlement patterns at the time meant that smaller towns and cities were usually only inhabited by one religious group anyway. Bigger cities like Jerusalem or Antioch were inhabited by all different kinds of religious and ethnic groups, but otherwise, people who followed different religions typically didn’t interact very much and didn’t really live together. Different types of eastern Christians might mingle together (Syrians, Greeks, Armenians, etc) and different kinds of Muslims might live together too, but even then they probably mostly lived in separate neighbourhoods, if not completely separate towns.

So when the crusaders arrived, they often came across Christian towns that were happy to see them. Sometimes it was a bigger town that had a Seljuk Turkic garrison, which might decide to flee instead of face a large crusader army (and a hostile Christian population within the walls). Especially in eastern Anatolia/northern Syria, where there was a huge Armenian population, some towns were happy to expel the Seljuks and replace them with crusaders. In 1097 the crusaders took control of Malatya and Tell Mannas this way, and the first of the crusader states, the County of Edessa, was founded when the crusader Baldwin of Boulogne married the daughter of Thoros, the Armenian ruler of the city.

Artah and Marash are a couple of other tows that the crusaders took peacefully in 1097 and 1098, in the north around Edessa and Antioch. In the south, when the crusaders were besieging Jerusalem in June of 1099, the crusader Tancred was invited to take over Bethlehem by the Christian inhabitants.

Sometimes Muslim towns surrendered peacefully as well. In April of 1099 the crusaders were besieging Arqa, which did resist, but they were also trying to find a suitable port along the coast that would help them with their siege. Raymond of Toulouse had a small force to the port of Tortosa, just north of Tripoli, but one night he lit a bunch of fires to make it look like a bigger army was there. The population fled the city and the crusaders took it without a fight. Not long after the port of Marqab surrendered without a fight too. Later in June, when the crusaders arrived at Jerusalem they found nearby Ramla already abandoned.

So cities certainly did open their doors to the crusaders - usually when they were already Christian, but sometimes Muslim cities chose not to fight too.

Sources:

Thomas Asbridge, The First Crusade: A New History (Oxford University Press, 2004) - my usual source for the events of the First Crusade

Ronnie Ellenblum, Frankish Rural Settlement in the Latin Kingdom of Jerusalem (Cambridge University Press, 1998) - for a bit of a deeper look at settlement patterns in the Near East, before and during the crusader period