Can I reuse the garage door rail when replacing the garage door opener?
Generally no, you cannot reuse the rail, they are not all exactly the same dimensions. There is no one standard size of profile, even within the same manufacturer, they change over the years. 

The holes for the attach points may change also, this could be solved by drilling more holes, but this would defeat the purpose of saving time.

If you bought a new opener within a few years of buying the old one, then maybe. But, they normally last 10 years or so. 

Edit: just to be clear, this is the rail on a typical opener.



For most designs, a shuttle rides on the rail.



Not all rails have the same cross-section (profile). There are at least two types, T and Box. And each of these are not standardized.



You could possibly re-use the shuttle and possibly the chain. But, these are high wear items, not recommended. Not all use chains anyway, there are belt drives also.