I have had and made hundreds of dyneema things like yours, the rings will not get hot enough to damage the line.
The only real problem is a running line rubbing against the dyneema, just a bit of awareness when setting up rigging etc.
One thing to remember is that dyneema has a low melting point but has an extremely high heat absorption point, that is why it's used in bullet proof vests, so it takes a great deal of constant heat to fully sheer it.