I have read that if you have tattoos you are not allowed in hot springs (onsens)! Is this true?

. Yes & No

Yes this varies a lot depending on the place. Some places will absolutly refuse you. Others may ask you to cover it. Again others may say you should not but will do nothing to enforce it.

Size does matter if it is just a small tatoo you shouldn't have any trouble. If you have a full body tatoo, or large tattoos. I am sure you will be asked more then not to leave. 

I have a small tatoo on my arm and have never been asked too leave. Although I have yet to try at an expensive onsen. One gym insisted that I cover my tatoo while working out.

So it varies from place to place.

Just remember though this is Japan and the culture is different. So respect the culture. If you are asked to leave don't make a scene . Just apoligize and find a new place to go.  If you can't do this then Japan is the wrong country to be living in.