Hi Jessie, there's some debate about whether MS can cause heartburn or not. Nobody really knows but a lot of people on here seem to suffer from it and being that MS seems to affect just about everything it does seem possible. As far as I know there's never been any studies on it.
There are two other possibilities... one is 'MS hug'. (google it if you haven't heard of it or look on 'symptoms' on this site). That can certainly feel like a pain in the chest sometimes and I've had it so it's very painful.. The other is a spasm of the oesophagus. That can be caused by MS although thankfully not common.
Really glad you're seeing the GP as you need to get it sorted. Jessie, just to be on the safe side, if you get the pain and feel dizzy, sick and weak at same time, you need to call an ambulance or get yourself to A&E. Sorry love don't mean to frighten you and I'm sure it's to do with Hug, spasm or indigestion... but always better safe than sorry. I called ambulance once and turned out not to be an emergency (was side-effect of a drug) and they were really nice about it... so don't worry about that.
Anyway, yes you must tell the GP.
Hope you get it sorted soon,