Repro in debugger.

This appears to be two seperate issues.

LiveUnwrapPinFail.blend has three islands and only one seam. When Live Unwrap is enabled, all three islands are merged into one island. During the merge, some of the pinned UVs are forced to be identical, and the number of pinned verts drops from 6 down to 4. After this merge occurs, the feature seems to be working as designed, even though it is unlikely what the user originally intended.
As a work around, the user can select "Seams From Islands" before activating Live Unwrap.
More discussion is required for a correct fix in this situation.

For the second unrelated issue, LiveUnwrapCube.blend, the cube has two pinned vertices and no boundaries or seams. The current unwrap solver cannot unwrap a mesh like this because the problem is "underconstrained", so instead it ignores the mesh completely.
In this circumstance, it might be possible to add additional constraints to the solver to get a solve. I will investigate this solution to see if it is possible.
In the mean time, the user can add or remove a pin, or, (by co-incidence) adding a seam or boundary by using "Seams From Islands" will also avoid this issue.

Update on the second issue, LiveUnwrapCube.blend, with two pinned vertices and no boundaries or seams.

This is the original UV layout:

Live unwrap currently excludes these charts (i.e. UV-Islands) as they have no boundary. This is the current behavior, from a UI perspective, it appears to the user like the "Live-Unwrap" is not changing the interaction.

If we remove this limitation, we obtain the following degenerate result:

This next proof of concept code change shows it is possible to get a non-degenerate result by adding additional constrains to the LSCM solver:

Note that this is not a "good" unwrapping, it's simply an existence proof to show that the LSCM solver can still produce an output given this set of inputs by adding additional constraints to push against.

One option would be to find additional constraints which produce an unwrapping which is "useful", even if the mapping itself is not "good".

Another option might be to special case the "2 - pin" case in the same we special case the single-pin case. For example, we could generalise the single-pin case to perform a rotation like so:

A third option might be to attempt to temporarily add additional (boundary) constraints, but this might not work for all inputs.

There might be more options for a fix too..

Are there any technical artists want to chip in with their preferred interaction here?

*Edit: Grammar

Checking: LiveUnwrapPinFail.blend as far as I can see, unwrapping in the UV editor should calculate seams based on existing islands, running "seams from islands" then unwrapping works as expected here.

LiveUnwrapPinFail.blend is fixed by e6e9f1ac5a2dae42dc523

LiveUnwrapCube.blend is now T98240