It's funny that both heartburn and sinus issues are coming up here. Both for me were resolved through diet changes. I was taking a ton of meds for both heartburn GERD and sinus issues. I'd already had two upper endoscopies.
When I went strictly Paleo in 2012 I cut out all the inflammatory food groups. I was pretty strictly low carb Paleo for about a year. I added some things back in gradually like rice, dairy, and potatos/nightshades and none bothered me. I still stay moderate to low carb and still maintain a gluten/wheat free diet, because the consequences of not really suck.
Now, unless I get accidently glutened or I overdo it on non wheat sweets, I have no issues with heartburn at all.