Can local anesthetia cause toe pain?

It is possible: A nerve may have been nicked creating discomfort. It may last weeks or several months. Ice might reduce the discomfort.

Yes...: Any injection with a needle can cause pain.

No: This is being caused by something else.

Not directly: You don't say where the local anesthesia was administered. If it was for a digital block of the toe, then i imagine it would be possible to injure a nerve during the injection. Otherwise no.

Not for most: The injection itself burns but most get no long lasting pain unless a nerve was damaged but chances of this are very low.