Looking to get a Primary Arms 3x scope or a Vortex 3x for my AR. I have the standard front sight post. If I want the scope to clear that how tall of a riser do I need to mount the scope on it?

If your scope is more than 3x, you will not be able to see the front sight as it will be inside the minimum focal distance of the scope (regardless of the height of your rings).

Ring height for scopes on ARs is more of an issue of establishing a comfortable check weld. For a gun like an AR that has a "straight comb" stock, you're looking at high or extra high rings.

But with an AR, you're pretty lucky because you can just get an AR-specific one piece base/ring combo like the offerings from ADM or LaRue, or even the Nikon PEPR if you want something more budget.

Oh excellent.

What about a quick flip mount so I could use my BUIS?

There is no mount that I know of that flips your optics to the side (like you see with magnifiers).

You can QD the optic by taking it completely off, then use your irons. Just mark your mount and receiver so when you put your optic back on, it will return to zero.

To amplify what LDD pointed out, tip-off or flip-over mounts for scopesights were somewhat common long ago, but they don’t seem to exist these days. It might be possible to make a mount intended for a magnifier to work, but the inherent problem with any mount that allows the sight to move around is the potential for poor accuracy and loss of zero.

If someone believes it might be necessary to dispense with his optic and resort to a backup sight in the heat of a firefight, a quick detach mount is the obvious solution. In my experience they don’t necessarily return to exact zero, but presumably that wouldn’t be the shooter’s primary concern in such a situation—especially if something happened to make the optical sight unusable in the first place. Taking a bullet strike to the objective lens will probably mess up the zero pretty bad even if the mount is undamaged.

confused by this post Voshterkoff. Don't see that either of the optics he proposed are compatible with this mount.
But in any case the two optics listed in the initial post (I would never use them) appear to come with mounts.