Some, like Conor Friedersdorf and Kevin Williamson, have problems with the no-fly list, saying it penalizes people without any legal process or chance of redress. There's also the FBI terrorist watch list, which apparently overlaps no-fly but is different. In a different area, we have the sex offender lists. IMO all three lists deprive people of abilities they'd normally have. The right says that denying guns to people on such list is denying them their Second Amendment, Constitutional right, which is wrong.
I think Friedersdorf and Williamson have a point: there should be a legal process for review and possible challenge when people lose, possibly for the duration of their life, some abilities. I think that's true even for sex offenders, who have already gone through a legal process. People can grow and change, people can be convicted in error.
The Senate is to vote on the issue in this coming week--several proposals, none of which are likely to pass. I've not studied the issue, but I think, provided there's a review process at some point, it's reasonable to deny guns to those on those lists.
And having said that, I don't think such a restriction would do much to avert mass shootings. Even Mateen would have passed that test, since he wasn't on the FBI list when he bought his guns.
I like the New York gun laws, including the requirement for friends to sign onto the application for a permit, but even with those laws Jiverly Antares Wong killed 13 people just a few miles south of where I grew up.