Is there a food that resembles the texture and consistency of ice cream but is not cold/does not have to be kept cold?
Much of the distinctive experience of ice cream comes from its temperature, so be prepared for disappointment. However, a set custard like in a crème brûlée or panna cotta is creamy and holds its shape, and can be flavoured as ice cream can be. Alternatively, a mousse has air bubbles incorporated into the mixture so is much lighter, although less creamy.
Because it cannot rely on ice crystals for solidity, it will necessarily be richer than ice cream so you'd want a smaller quantity (I'm imagining an ice cream cone filled with custard and it is way more than I would want to consume).
The consistency is light and airy due to meringue, so not really the same as ice cream but perhaps close enough to get the name.
I would argue for a clear no - it is impossible for such a thing to exist. It is possible to think of dishes that are for some reason "related" to ice cream - but if you serve them to ordinary people, nobody would take a bite and spontaneously say "wow, this is just like eating ice cream, only warm".
The reason behind this is less related to cooking and more to the way people conceptualize the world around them. We tend to lump similar things into categories. And it so happens that we don't categorize ice cream by its texture, but by its iciness. If you look closely, the dishes which we call "ice cream" have a ton of different textures. There are ice creams with eggs and without; sorbets; sherbets; granitas; gelattos; dondurmas; vegan imitations of dairy ice cream; there is the foamy stuff you get in the supermarket, the dense rolls from the insta-cold plate, the gelateria ice cream from the small compressor machines, and the icy sorbet that kinda failed when you tried the ziplock method, but you still keep in the back of the freezer. Specialists may give them different names, but people tend to simply think of them as "ice cream" - because the most salient feature of ice cream is not the texture or the ingredients, but the fact that it is frozen-and-scoopable.
Now, one could argue that there are other important factors for something to be considered ice cream, such as being sweet. And if savory ice cream is readily perceived as ice cream, then warm ice cream should also be perceived as ice cream. There is some logic to this proposition, but it doesn't hold in reality. People are very accustomed to having sweet, scoopable, but not frozen desserts (some of which are listed in the other answers), and they just don't associate them with ice cream. I have even had ice cream recipes where the base is not liquid, but firm enough to be scoopable - and yet, it was clearly a "cream" (in the texture sense, not the dairy sense) but not an ice cream. To have something recognized as "ice cream", it really has to be icy cold.
It's subjective of course, but one could say that fudge has something like the texture and mouth feel of ice cream. Some kinds of cake icing could be said to have a similar texture and consistency as well.
I conjecture that this is because the texture of ice cream comes from tiny ice crystals, while these foods have a texture that comes from tiny sugar crystals. In both cases the crystals melt in your mouth after you take a bite.
Yes, it is possible - using techniques from molecular gastronomy
Heston Blumental claims to have managed to create a hot ice cream in 2007, but there are no further details on that I can find. For a demonstration of his hot sorbet though, which would presumably be made using similar techniques, check out this youtube video.
Broadly, the technique involves creating a 'fluid gel' using hydrocolloids known as gellan, which remain stable under heat. Alternatively, it can be made using methylcellose, which has the added amusing property that it will "melt" as it cools.
Disclaimer: I have not tried to recreate these recipes in my own kitchen, nor have I tasted the results on them so cannot comment on how closely they mimic the mouthfeel and/or taste of icecream.
This is not an answer to the question, but a long comment about a thing that might get tangentially related and someone might think about.
Although not for ice cream, there is "Hot Ice" [Wikipedia]
Hot ice, or a supersaturated mixture of sodium acetate with water [youtube] , is a liquid, that, when touched rapidly crystallizes and releases heat and creates an effect similar to ice.
It is also with noting that sodium acetate is used in food processing with acetic acid as E262 to give potato chips "salt and vinegar flavour". I highly doubt it would be good for ice cream, though. :)