He's barely played since 2018. He looked like our best option beyond Sexton before the injury problems started, and he needs game time to play himself back into that kind of form. If it was me and he's fit, I'd start him.
I like Burns, both because I'm an Ulster fan and because I like his style of play. He's a risk-taker with an exciting attacking kicking game. But I'm not sure that's the game Faz wants to play - Joe Schmidt certainly wouldn't have entertained it - and while Billy can play that game at Pro14 level, international rugby is a big step up, and I'm not sure he has the big-game mentality. He also benefits for Ulster from having John Cooney in front of him - defences have to watch his running game, which makes more space for an out-half, and his decision-making and leadership also take a lot of the pressure off the 10. For Ireland he'd be playing behind Gibson-Park, who is a much more basic scrum-half - he doesn't do a lot wrong, but he's a lot less pro-active than someone like Cooney, so the responsibility and decision-making would fall more on the 10 (although playing Burns behind Caolin Blade might work). Cooney being Ulster's first-choice goal kicker also means Billy lacks match practice at that.
If they do pick Harry, I'd at least get to see something of him, see if he's as good as his reputation. The only time recently I thought I'd get a look at him for Leinster, he got injured in the warmup.
And If they want to be really experimental, Michael Lowry's played a few games at 10 for Ulster!