Remember: I'm not saying you can't stay airside. I don't know. It seems that LIR is not set up for int'l-to-int'l transfers, but I could be wrong. Do report back how it works.
So, assuming you have to go through passport control ...
If MIA-LIR is on time:
That bodes well for you, especially if you're in the front of the plane. That is the first incoming international flight of the day, so you won't be caught in an immigration bottleneck. You will be going through security, though, with your fellow LIR-DFW passengers as well as any dawdlers on the LIR-MIA flight going out on the plane you came in on.
If MIA-LIR runs late:
You will get bunched in with other international arrivals, in particular, DL's ATL-LIR flight getting in soon after your scheduled arrival time. A late arrival will cut into your time for formalities, of course. A too-late arrival will be a problem in that you're not going out on the same plane you came in on.