Why closer? Gravity is a measure of acceleration, not force - replace the moon with an Earth-sized planet and it would accelerate towards us at exactly the same speed.

Of course, we'd accelerate towards it about 50x faster than towards the moon, which would rapidly destabilize things, but as long as we gave Earth a good strong sideways kick at the same time so they didn't collide on the first pass, the two should orbit their mutual center of mass, just as we currently do with the moon - even if that center is currently 1700km below Earth's surface. And over time tidal energy transfer would circularize the orbits, and tidally lock us to each other. Assuming we didn't give just the right kick to start with.

Is there a minimum stable distance for tidally locked binary planets?

They can't escape without one getting a huge outward kick of energy, and they can't collide without a huge inward kick. Tidal energy transfer is no longer happening, and I think you have to be pushing spiraling neutron star densities and speeds before you can shed much energy through gravitational waves. And so long as the sun is many orders of magnitude more massive, you're not going to get any pesky chaotically complex three-body problems cropping up. Just don't try to add a moon.

I suppose if they cross each other's Roche limit, where tidal stress will tear moons apart into rings, things might get complicated... but a quick search says that's only ~11,000km for Earth - a twin planet would actually be touching the surface before its center of mass crossed the line, but I think even such a "dumbbell planet" would still be stable so long as it wasn't spinning fast enough to throw stuff off the surface at the outer tips (you *know* that's where the spaceports would be!)