Tomato sprouts generally develop from planted tomato seeds or from tomatoes that have decayed and released their seeds. It is not uncommon, however, to find tomato sprouts growing inside an intact tomato. Usually the gel around the tomato seeds inhibits any sprouting while the tomato is intact or even after the tomato has fallen to the ground and has decayed. The seeds are allowed to sprout only when conditions are favorable for growing tomatoes; for instance, the seeds will sprout in spring and not in winter. Sometimes, however, temperature fluctuations can lead to sprouts inside a tomato.
These tomato sprouts, along with tomato leaves and stems, have, for most part, been eschewed in cooking. Some cooks have successfully used them to add a distinctive tomato flavor to sauces and some other tomato dishes, but it should be noted that, in these cases, the stems and leaves are simmered for flavoring and are then removed; they are not eaten. The reason for this is that tomatoes belong to the nightshade family of plants. The leaves, stems and unripe fruits of many plants from this family are known to have toxic properties, and can cause health problems if they are eaten.
A glyco-alkaloid known as tomatine is present in unripe tomatoes as well as in the leaves and stem of the tomato plant. The tomatine alkaloid has been known to adversely affect the nervous system and to cause irritation of the gastric intestine. Some people also get allergic reactions by merely touching the tomato leaves, and, in the case of animals, particularly dogs, eating tomato plants can sometimes prove fatal.
On the other hand, unripe tomatoes have long been used in many food recipes without anyone coming to harm. Many people have also eaten sprouted tomatoes without any apparent ill effect. Lab experiments with animals have shown that tomatine is not absorbed by the body, but is expelled along with unwanted cholesterol. Tomatine has also been found to have strong anti-microbial properties and to have a deterrent effect on the growth of cancer cells.
It seems then that in small amounts, tomatine is safe to eat. It would take a large quantity of tomato sprouts, leaves and stems to give a human a toxic dose; eating one or two is not likely to cause any kind of health issues. Still, if there are any doubts or if there are any allergy issues, it might be advisable to avoid eating sprouted tomatoes.