There are four main problems I see with planting seeds from grocery store produce.
- They might be genetically engineered, unless they're certified organic.
- They might be patented or protected by plant breeders rights of some kind. If they're genetically engineered, odds are extremely high that they are patented, unless they're over 20 years old.
- You really don't know what they are, because even when they label a tomato Roma, for instance, that doesn't mean it's necessarily a Roma tomato.
- It could be illegal, depending on the form of produce and where you live, due to state or other laws.
I recommend contacting the grower to find more information about whatever you'd like to grow, but try not to let them know you're wanting to grow them, and either way, don't expect a useful, thorough answer.
For people who are selling fruits marketed as heirlooms, you're really just trusting them to be selling what they say they are. Ask them what variety they are. They probably are heirlooms, but if you're going to grow them you'll probably want to know the variety. Lacking the variety, you'll want to know where they were grown to make sure they'll grow in your climate.
GMO seeds should normally be as fertile as heirlooms, and they'll likely breed true. However, since they're probably patented they might cross with your other stuff and give you serious legal problems that I don't recommend you want to deal with. Also, they may produce Bt toxin and/or such (e.g. Bt corn). Bt is not botulism toxin (I state this because one person who suggested an edit thought that I meant this), but rather it's a fairly harmless toxin (to humans) by comparison, and it isn't produced by the same kind of bacteria. Plants, such as corn have been purposefully genetically modified to produce Bt toxin to keep insects away, as Bt toxin has an insecticidal influence.
Hybrids are often fertile, depending on what kinds of plants you're talking about, but it's very statistically unlikely that they'll breed 100% true. It helps if you know the parentage of your hybrids. Then you'll be likely to know if they had differing numbers of chromosomes or such, or if there are any definite undesirable traits from parents that might show up in the F2+ generations.
It's also good to know that some forms of grocery store produce in the form of bulbs, tubers or such, constitute a misdemeanor to plant in some states (e.g. potatoes, garlic, and onions in Idaho), unless maybe they were grown in the state they're being sold in (but don't quote me on that last), or unless they're certified by a certain process or some such (which grocery store produce is very unlikely to be). The idea seems to be to reduce the spread of diseases.
Grocery store produce is often capable of growing, but it may (or may not) be diseased or weakened by chemicals, it may (or may not) be true to type, it may not be used to a climate like yours, it may be genetically modified, and it may have legal and ethical issues. In summary, you can probably do it, depending on what it is, but I don't recommend it at all in this day and age (unless maybe you get it organic from a farmer's market, and actually know the breed). You'll most likely have a more enjoyable experience if you get forms that are intended to be grown.
It is a bigger issue when people who save seeds to share and/or who breed plants grow plants from grocery store produce, because they might be illegal, and they might cross-pollinate their other stuff (and/or their neighbor's stuff), or be used in their breeding programs (and then they might share the seeds with or sell them to people who don't know they're illegal and/or of questionable origin). Growing them is one thing (still a questionable and potentially illegal thing, though), but allowing them to potentially or purposefully cross with other crops is another.