First thing to ask: What kind of overload are we talking about?
High bandwidth alone? This in turn can either mean that your site is more popular than expected-- which you said isn't the case-- or that somebody is using it as a proxy to get at some other site(s).
Excessive computing resources? This can happen if a site is much more php-intensive or database-intensive than earlier sites you've had.
when I hit the IP in the browser it lands on one of my other sites on the server
Did you mean, the offending site's IP? Not any of your own? Then you get to the possibility of a DNS problem.
Does the offending IP belong to the same host? Doesn't have to be the same physical server, just the same owner somewhere alone the line.
the access log shows my server / IP accessing the file right? If it was in a loop it wouldn't keep making new requests, would it?
This is obscure. Do you mean that this is what the log actually does show-- or what the log would show under such-and-such circumstances?
An infinite-redirect loop that is intercepted by the browser will show up in logs as a series of identical requests, each met by a 301 response. The exact number is up to the browser; last time I tested (one human request, one browser) there were ten requests. A 301 by itself is the smallest kind of response, since there's no page being served at all. But if there are lots of them, the request itself could overload the server.