IANAD, but I have actually had lingual nerve damage (paresthesia) from a dental injection.
1. This is very rare. The endodontist who gave me this injection said it has happened to only 3-4 of his patients in decades of practice, and I assume that endodontists probably do this injection more often than general dentists. You are probably more likely to get into a car accident on the way to the dentist office than to get this type of paresthesia.
2. Only one particular injection - the IAN block - is known to cause injectional paresthesias.
3. 95% of injectional lingual paresthesias are temporary, most resolving within a few weeks, some taking longer (mine lasted 4 months).
4. If you are really concerned about the risk, there are a few things you can do to reduce the risk:
a) ask the dentist to try an infiltration with articaine/septocaine first and see if that numbs the tooth enough. There are some dental journal articles that have found that this can be successful for mandibular first molars (#19, 30) in many cases, allowing you to avoid the IAN block.
b) ask the dentist to use 2% lidocaine instead of a 4% anesthetic for the IAN block (there is some possible evidence that the 4% solutions have a higher incidence of paresthesia)
5. If you don't get the filling replaced, and it leaks bacteria into the inside of the tooth, a lot of the tooth could be destroyed by decay, resulting in way more problems, pain, and dental work (including injections). I personally would not hesitate to have a leaking or damaged filling replaced even if it requires an IAN block, and I know exactly what I am risking.