I also have tennis elbow. It was brought on by too much indoor climbing but the MTBing probably didn't help. I haven't climbed since early last year and was seeing a physio for a while but it still isn't fixed. Here are a few things that have helped me.
1) Drop and the saddle and move it forward a bit and/or raise the bars and move them back to avoid leaning on the bars. It's also a better position for technical stuff.
2) Get some fat, sticky grips. That will help you to feel more confident and in control and it will then be easier to relax your grip on the bars which should help the elbow. This is also better technique. I like ODI Rogues.
3) Avoid yanking on the bars.
4) Avoid riding stiff hardtails on hard, bumpy ground.
Hope this helps.