Ceviche is basically using just lemon juice to "cook" the fish, so I was wondering if there are any fish varieties that wouldn't be recommended using this method?
In general, you want to avoid "oily" fish, because it will tend to go rancid rather than curing properly. By oily fish I'm referring to sardines, mackerel, bluefish, and similar "fishy" fish. Farmed salmon is also generally a bad idea because it's so fatty; in general, you would only make salmon ceviche if you had a specific recipe for it. Farmed trout and true sole also don't work very well, as they're too delicate and tend to fall apart into a fish mush while marinating. Finally, cartiligenous fish, such as shark, skates, and chilean sea bass will be horribly gristly and difficult to chew as ceviche -- and you shouldn't be eating them anyway, they're endangered!
Pretty much, you're looking for a lean mild-flavored white-fleshed fish: tilapia, flounder, rockfish, cod, John Dory, mahi-mahi, and similar fish will work in any "generic" ceviche recipe. There are specific recipes for tuna, swordfish, shrimp, squid and octopus which are also excellent, but only if your recipe calls for it.
My initial reaction was "stay away from meaty fish like tuna or swordfish" but a quick Google shows those as common variants. That said I like white fish, squid, scallops, and shrimp. But I don't think anything is strictly off limits.
There are also variations on ceviche that use gently poached fish instead of raw. If you're at all worried about the quality of your fish, you could try poaching it first.