It looks like you had the grass mulch up against the stem of your tomato plant. You should give the stem some space so it's not directly touching your mulch. In some of your pictures you can see roots starting to grow where the mulch was.
Based on what I'm seeing it looks like you spread the grass soon after cutting when it was still green. Green grass clippings are high in nitrogen and contain a high amount of moisture. When grass clippings start to decompose they can get quite hot. The high nitrogen and moisture may have damaged the stem as well as the high moisture, reduced air flow might have been a good environment for disease.
The first two pictures don't look too bad and I think the plants will be fine. The third picture looks like more damage to the stem and I think at the least the plant will suffer a little bit.
Pull back the grass from your existing plants and next year remember to leave space so the grass mulch isn't touching the stems. When I used to use grass clippings I would pile them up in one section of the garden after mowing then a few days later after they browned out a bit I would spread them around plants.
Regarding the heat generated by grass clippings this is a portion from Missouri Unversity's Extension Office. http://extension.missouri.edu/p/G6958
Yes, grass clippings used as a mulch should be built up gradually to a
1-inch layer using dry grass. Greater thickness can inhibit the
penetration of moisture and oxygen into the soil, and excessive heat
and foul odors may develop. Mulching thickness can be increased by
mixing in a 1:1 or 2:1 ratio of compost, dry leaves or wood chips with
fresh grass clippings.