Answer: I like to pick tomatoes when they are fully ripe because I think they taste better that way, but a nearly ripe tomato will continue to ripen after it is picked. If you pick a tomato too early it will either rot before it ripens or the flavor will be poor. Picking them a day or two early might conserve a small amount of the plant's energy, but probably not enough to make that much difference. Letting the tomatoes become over-ripe results in a spoiled harvest; if they are too ripe the tomatoes will fall off anyway.
While it is true that plants flower and fruit in order to reproduce themselves, failing to remove the "seed heads" will not always cause the plant to die. Instead, it may just cause it to stop producing. If it is an "annual" it will "go to seed" and die. If it is a perennial (or say, a tree or shrub), it will direct some energy to seed production rather than to plant growth, but that won't kill the plant. Flower gardeners regularly "dead head" or pick off spent blooms (seed heads) to keep their annuals blooming heavily all season; they groom their perennials for appearance and to conserve the plants' energy, although in some cases it may also extend the plants' blooming season somewhat.
Picking most vegetables regularly will encourage the plants to continue making more in an effort to reproduce. For example, bush beans will produce for a long time if the beans are kept picked before maturity (eventually the plants will exhaust themselves, however). This also explains why a cucumber or zucchini vine will stop making new cukes or zukes if you miss a mature one.
Tomatoes are interesting because there are two types, determinate and indeterminate. A determinate plant grows to a finite size and stops. Picking those tomatoes, ripe or unripe, will not cause it to start up again. Indeterminate tomatoes, however, are essentially perennial plants and theoretically could be kept growing for years under the right conditions, as in a greenhouse. In time I suppose they would decline in vigor and perhaps exhaust themselves, but in my garden they keep on blooming and making more tomatoes as long as they are healthy; there are tomatoes on the vines right up until frost kills them.
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