Answer: If the tomatoes have dark green shoulders and light green bodies they will ripen off the vine. If they are dark green all over, they probably won't ripen. Lots of fungal problems can attack tomato plants and usually by the time you recognize them, it's too late to save the plants.
Without seeing the plants I can't really diagnose the problem, but your plants could be suffering from either early or late blight, which are caused by different fungi. Early Blight is caused by the Alternaria fungus. It overwinters on infected plant material, even seeds, so it's hard to completely remove the spore reservoir from the garden by cleaning up all the vines and fruit. Early Blight works slowly, whereas Late Blight (caused by the Phytophthora fungus) may kill plants within a week. The fungus is always growing somewhere and releasing spores into the air, which moves on wind currents. It doesn't overwinter in your soil, so something like solarization won't help control it.
If you are growing tomatoes in the same site every year, the soil could use a break from constant monocropping, but if you don't have room for a second plot, I suggest loading up the soil with good compost. Compost contains lots of helpful organisms which can work against disease organisms. Work a few inches into the existing soil, and then spread several inches on top of the soil as a barrier mulch. Once your tomatoes are planted, mulch with straw as well. Keep your plants healthy so they'll be in top condition to resist disease. Stake or cage them, and keep the lowest leaves from coming in direct contact with the ground.
There are other disease and pest problems that may be affecting your crop, such as Verticillium wilt and Fusarium wilt. The above recommendations will also help with these diseases.
When choosing tomato varieties, look for the words "disease-resistant" or the letters V (verticillium), F (fusarium), N (nematodes), T (tobacco mosaic), and A (alternaria). Burpee's most disease-resistant varieties include Celebrity, SuperTasty, and Big Beef. Burpee's roma-type, 'Viva ItaliaHybrid', is resistant to Verticillium, Fusarium, Nematodes, and Bacterial Speck. As far as I know, there are no varieties resistant to the late blight fungus (Phytophthora).
If you follow the above practices every year, and choose resistant varieties, I'll bet the incidence of all disease will drop. Good Luck!
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