How do I Position an Element within Heading (H1, H2, etc) tags?

You have to put it first

<h3><input type="button" style="float: right">Order Details</h3>

That's not w3c compliant is it?

Can't you wrap the whole thing in a div,

<div><h3>Order Details</h3><input type="button" value="Refund" id="btnRefund"  /></div>

And apply your styles to the div, rather than the h3 if you need some form of uniform look? Float the h3 left or use display:inline so they appear side-by-side.