An F1 tomato hybrid occurs when you have two tomato plants of different varieties and then one is pollinated by the other variety's pollen. The seeds of the resulting cross-pollinated tomato will grow plants that are F1 hybrids. The children of F1 hybrids are F2. The children of F2 are F3, and so forth.
There is only one domestic tomato species (Solanum lycopersicum, AKA Lycopersicon lycopersicum, AKA Lycopersicon esculentum), but there are thousands of varieties within this species. As long as the parents are both pure domestic tomatoes, then the seeds should be fertile. It's like with dogs. As long as both parents are domestic dogs, the puppies should be fertile, even if their parents are of different breeds. They're all dogs.
However, there are several species of wild tomatoes that can cross with domestic tomatoes. This complicates things. It's possible that some wild tomato species would cross with domestic ones and have sterile offspring. However, I don't know that this is the case with any of the wild species, but it is possible. I'm guessing many, if not most, will still be fertile. I'm pretty sure at least Solanum cheesmaniae and Solanum pimpinellifolium could have fertile offspring with domestic tomatoes; probably Solanum galapagense, too.
There are a number of tomato varieties that are part domestic tomato and part wild tomato. For instance, Indigo Rose (a PVP protected variety) has some Solanum cheesmaniae in its history. I don't suppose it would be easy to determine which tomatoes are purely domestic anymore. But, odds are in your favor that it'll probably be fertile.
So, to answer your question, F1 tomatoes are usually fertile (but possibly on rare occasions, a sterile one might exist, although I've never heard of one, so far). However, hybrids don't usually breed true. That means that successive generations are not necessarily exactly like the previous one, because they have variable genetics (like mongrel dogs do). For instance, purebred dogs breed the same kind of dog. Mongrel dogs breed all kinds of mongrel dogs and few of them (comparatively speaking) look the same.
(I personally like the idea of mongrel tomatoes, as long as I know what the parents are and none of them are patented or protected varieties, and I like the possible combinations they might produce.)
It's worth noting that an F1 hybrid where both parents are pure domestic tomatoes is likely to be exactly the same every time that same kind of F1 hybrid is bred. I mean, if I cross a German Pink with a Black Plum, I'll get the same kind of tomato you would get if you crossed a German Pink with a Black Plum tomato the same way. Which parent is the mother and which is the father does seem to matter. Crossing a German Pink with a Black Plum is not the same as crossing a Black Plum with a German Pink tomato. mtDNA, cpDNA, and similar kinds of DNA generally only come from the mother plant (not from the father plant), I believe.
Hopefully this clarifies.
If you want to grow mongrel tomatoes, I recommend hybridizing them yourself so you know what's in their genetics. Planting seeds of commercialized hybrids is a bigger risk because they may not care about what the F2s will be like, and they rarely say what the parents were.