I don't think the subjunctive applies here. I think it should be "continue" because a plural subject needs a plural verb. Say you eliminate the word "there", which is just a device to move the subject of a sentence to the back and, to simplify things further, take out the word "any". Your choice then becomes between "If problems continue�." and "If problems continues�."
The first looks right so let's put "any" back in. Now you get "If any problems continue�." Add in "There" and "to be" and move "problems" to behind the verbs. "If there continue to be any problems�."
A parallel example would be "If there are further problems". If you wanted to use "is" you would say "If there is a further problem" . Also correct are versions using the subjunctive: "If there should be further problems" or "If there should be a further problem". But who would now use those in an email? (Except me, obviously)
However, I agree with Quizmonster that few people will have a problem with your version. Look at how the votes went both ways in this thread. Much as I would like English to stay with the rules I was taught as a child, it doesn't. It's a living language and democratic to boot.