I have never tested this, but here are my thoughts anyway.
There are 2 components to ISSU.
First, there is the hardware component. All of the FPCs, PICs or DPCs on your router should support ISSU. If not, they will be reset when you switchover from one RE to the other RE (which has a different code). Assuming your Logical Systems to be implemented on ISSU supported hardware, you should be fine. If not, you would suffer from the same effect outside of LS as well.
Second, there is the software component. Obviously the kernel and so on must support it, but more visible is the protocol support. You need to make sure you can switchover from one RE towards the other without losing adjacencies, peerings, .... , so basically neighboring routers can't tell you've switched. This is a hard thing to do (especially the corner cases, ie when you switchover in an unstable network). In JUNOS, we do this through Non Stop Routing (NSR). Today, we're supporting close to all protocols (you'll find them in the manuals).
What is not supported is the fact that these protocols could be running from a separate LS. A protocol that is not supported will break up its neighborships as it will loose synchronization during the switchover (ie TCP windows not be corresponding).
So the effect of the latter in the LS will depend on the protocols that you use. If you have an LS with static routes, I would imagine the impact is minimal. If you take protocols like an IGP, BGP, PIM, .... you will also be depending on your neighbors on how fast they can build the new neighborship.
But a reboot is definitely not what I'm expecting. Your LS routing table will be impacted though and hence also your forwarding for some time.