Hi, Karen. Yes, there is a type of varnish that produces very harmful fumes as it cures/dries but, basically, you should have no concern about this as you cannot buy them anyway. These are called conversion varnishes and they are not generally available in retail outlets. They are only meant to be used in spray guns and, since they have a short pot life, are not available in spray cans or anything like that.
If you have further questions, I suggest that you consult Bob Flexnor's book, Understanding Wood Finishing. Since the book is geared toward the amateur or low volume professional, he does not devote much space to them. However, his book is excellent for the audience for which it was written.
Varnishes like you can readily buy use solvents like paint thinner/mineral spirits, kerosene or a distillate between the 2. These are relatively safe but you should read, understand and follow the directions and safety warnings on the can of the product that you buy.
It seems like there are people that have bad reactions to just about anything but the potential fire hazard in using the varnishes that you can readily buy is probably greater than the possibility of you having a bad reaction to the chemicals.
Bob Flexnor's book is available on line and from major booksellers. I use mine all the time and am happy to recommend it.