There are several types of solid fuel tablets but to answer the questions, they wouldn't be healthy to ingest, nor are their flames "clean" burning enough to recommend using as direct flame cooking. The most commonly used backpacking ones often smell like soured seafood and can occasionally leave that smell on the cookpots.
Handling wise, they are probably far less toxic than a small splash of gasoline on your hands from filling the mower. Technically not good for you, but it's a miniscule amount with limited exposure.
My take on it is wash your hands or try to pop them out of their package without handling them at all is best but I won't worry if I have to handle them a bit as in solid form they aren't absorbed into the skin as easily.