Are cells living by the time you eat them? I'm guessing for very cooked foods they're quite dead, but perhaps (some or all of) the cells constituting certain other food sources are alive by the time they reach the mouth?
If you eat fresh vegetables and/or fruits, then the cells certainly can be alive. Every time you eat a fresh (i.e. not cooked) salad or similar you are eating living things.
Almost all plant parts will survive quite a period out of the soil and retain "freshness", which indicates that they are alive - so much so that you can take a cut stem or even a leaf, place it in water and have it generate a whole new plant. This is known as vegetative propagation.
Regarding sushi, some sushi is served fresh as in, killed right in front of you. That tissue is certain still viable.
Regarding things like beef, that's usually aged several days before it's even butchered, and then often aged some more. That's not usually still viable tissue.
Obviously veg that has been served fresh is still living tissue, since depending on what you have you could clone or culture a new plant from the tissue samples.
Then there are live cultures (that have not been pasteurized); Pickles, Kimchi, Sour Kraut, Yogurt, Sour Cream, Cultured Buttermilk. Yes you are consuming live lacto bacillus or similar bacteria in that case.