I have a normal UK type “cold” roof, with interlocking tiles over the roofing felt. There are a few small rips in the felt about 5cm or 10cm long; I rather not have to get all the tiles removed to repair them. As the tiles are in a good state, I don’t have any leaks at present, but I rather fix the problem before it becomes an issue.
You can purchase self adhesive flashing which, in small pieces, should adhere well to the felt, and should be suitable for the heat and cold experienced as roofing underlayment. The rips are small enough that this should be sufficient - if you run into a larger rip you may need to choose a different method, or support the flashing as it may fall off due to its weight if it's larger than a few square inches.
Alternately, get more roofing felt and a tube of roofing caulk, and glue felt patches over the rips with the caulk. Make sure the caulk is rated for the temperatures, and apply it liberally so you don't create a water pocket.