Ignoring exceptions which occur in a 'finally' block is generally a bad idea unless one knows what those exceptions will be and what conditions they will represent. In the normal
try/finally usage pattern, the
try block places things into a state the outside code won't be expecting, and the
finally block restores those things' state to what the outside code expects. Outside code which catches an exception will generally expect that, despite the exception, everything has been restored to a
normal state. For example, suppose some code starts a transaction and then tries to add two records; the "finally" block performs a "rollback if not committed" operation. A caller might be prepared for an exception to occur during the execution of the second "add" operation, and may expect that if it catches such an exception, the database will be in the state it was before either operation was attempted. If, however, a second exception occurs during the rollback, bad things could happen if the caller makes any assumptions about the database state. The rollback failure represents a major crisis--one which should not be caught by code expecting a mere "Failed to add record" exception.
My personal inclination would be to have a finally method catch exceptions that occur and wrap them in a "CleanupFailedException", recognizing that such failure represents a major problem and such an exception should not be caught lightly.