I've never used royal icing. I have made a phrase from royal icing that is going on the cake I am now working on. The letters are red. Here's my question...will the red run when I place the letters on the buttercream?  Thanks in advance. 

I've never had a problem with my royal icing flowers running on my buttercream covered cakes.

Your red royal icing needs to be perfectly dry before you put it on the cake.  At least 24 hours.

You won't get a runny line, you will get a shadow of colour right under the letters after they sit on the buttercream for a day or more.

If you are still not sure, use a thin line of white royal icing first.  Let that dry for an hour, then pipe your red letters on top.  What will touch the actual cake is the white.