I stopped putting tomatoes in my compost for this reason, but the occasional one still makes it in. I grow both hybrid and heirloom maters, so I never know what a volunteer will be, nor whether or not it would yield anything worthwhile. And I don't have the veggie garden space to experiment on them. I toss 'em back in the compost pile. ;-)

My volunteer tomatoes were always cherry tomatoes.

I've got five volunteer potato plants going. Two in a pile of pre-compost, the other three in the garden.

I have had tomato seeds germinate and start to grow in my compost piles many times over the years and do not see any reason to not put them in my compost even today. Those newly germinated plants are easily incorporated into the compost, putting nutrients back, as well as adding nutrients fomr the seeds that do not germinate. I have also had squash, cucumbers, melons, many flowers, even dandelion seeds germinate and all, if not planted in the garden were quite easily reincorporated into the compost by turning the material over.
Aside from something being poisonous, such as Poison Ivy, Oak, Sumac, I see no good reason to not put potential sources of nutrients into the compost pile. I do compost the Pokeweed that I pull up around here.

I always have volunteer tomatoes in my garden area. They are welcome as replacements to the small tomato plants which get Early blight or eaten by deer in May thru July.

The volunteer plants may be small in comparison to the normal transplants, but they are usually hearty. They can be a surprise, since my normal plants are both hybrids and heirlooms. Since they are hiding among the weeds and grass, they tend to have some blight which needs to be cut off as soon as possible.

Some years I have dozens of volunteer tomatoes. Those I don't need to fill in spaces where the regular tomato was eaten or diseased are given away.

Yep. I've got volunteer tomatoes and something in the squash family. Why not?

I wish I could get tomatoes from my compost. It never happened to me. I have never seen a volunteer tomato, but I compost tomatoes from salads.

I hate to spoil the fun, really I do, but there is a very good reason not to use volunteer seedlings. They can carry or encourage diseases, specifically viruses, that have no effective controls. I have personal experience with this and went for years without being able to grow any type of squash because I had a virus in my soil. All I could do is wait and hope it went away. Eventually, I established a new bed and have been careful not to transfer soil from the contaminated beds into it and I have my zucchini and pumpkins once again. Most likely the virus came from a volunteer seedling that I let grow in the bed that then became contaminated. But don't take my word for it ...

For tomatoes:
"Destruction of volunteer plants is an important practice in the control of several diseases, because it prevents large populations of pathogens from surviving from one cop to another. Crop rotation also works to prevent crop to crop survival of specific tomato pathogens."

https://edis.ifas.ufl.edu/vh056

And for the squash family:
"Destroy volunteer cucurbit plants. These can harbor cucurbit disease organisms."

http://www.ag.ndsu.edu/pubs/plantsci/hortcrop/pp656w.htm

Understanding how diseases are transfered as well as where they come from is necessary to prevent the spread of them. While any plant might harbor a disease the most common means of spreading them is through insect pests, not the volunteers that grow in compost.
In some 50 years of gardening I have not seen any plant disease spread via volunteer plants that grew from seed in a compost pile. There is that possibility, although I think smaller then some think.

Nevertheless, it is a recommended best practice for preventing the spread of disease and, along with crop rotation, part of Integrated Pest Management.

I was wondering what to do with the the now more than 80 (!!) volunteer tomato plants that have popped up after I spread compost in the garden. I've no idea what kind(s) they are -- last year I grew only cherry tomatoes from packaged seed and none of them did terribly well (sandy soil that bakes up like concrete in summer here) but composted lots of store-bought (organic) large tomatoes and tomato bitlets. The 'volunteers' are looking a hundred times bigger, stronger, and more robust than the 6 cherry tomatoes I started indoors and have transplanted out. I'm going to stick those volunteers into every sunny nook and cranny in the yard and see what happens and what we get!

I don't compost my tomato plants. But each year I have dozens of volunteer tomato plants spring up from the fruit of last year.

At first I disliked them appearing rather late in the spring. But since there are always a few of my basement plants that develop some kind of blight or wilt in May, I now move some volunteers into the empty spaces.

Lived years ago in Aurora, CO. Neighbor got many truckloads of sewage sludge, topdressed his entire front yard several inches deep. Before he could lay sod, the area was completely covered in tomato seedlings, many thousands of them.