I'm not sure if I get heat rash in particular or not (but sounds similar), and I think what I get may have started before the Mirena... (For years I thought I was getting recurrent external yeast infections, and yeast infection meds didn't work!)
But I can tell you what has successfully stopped mine. Right before I go to bed every night, I use NeemAura Naturals body lotion on the area. I don't know exactly why it works, but I'm surely not going to stop because it has worked very well for about 2 years now. It's got neem and aloe in it, but I also think it's important that it's all natural, doesn't have things like mineral oil in it, etc.

Here are the ingredients if you're interested:
Ingredients: Neem Aura moisturizing complex (aloe vera 73%*, and neem leaf*,
goldenseal root, chamomile leaf, barberry plant, thyme leaf and ivy leaf
extracts), rice bran oil, stearic acid (from coconut), vegetable glycerin,
sodium borate (borax), decyl glucoside (from coconut), neem oil, jojoba oil,
allantain (from comfrey), tocopheryl acetate (vitamin E), dL panthenol
(pro-vitamin B5), lemongrass oil, chamomile oil, ginger oil, papaya oil, citrus
aurantium dulcis seed extract, citric acid (from citrus), xanthan gum, wheat
protein, natural fragrances, and nothin else.
*organically grown in accordance with the california organic foods act of 1990

I had itching from the time it went in until it was removed. It was crazy ! I was on a bunch of different meds. Nystatin, Diflucan plus OTC and nothing helped ! This was over an 11 month period. Since its been gone I havent had one bout of itching.

I have had awful itching as well, but since it started this winter I just atrributed it to dry skin and tried lotion but never really knocked it out. I never even thought that it could be related to the mirena. I am getting mine out in less thatn 3 weeks. I can't wait!