I am a US citizen living temporarily in Thailand.  There is a church
in the USA that wants to send some money to help a Thai church group
that is building a church in Thailand.  For reasons of funds transfer
safety they want to send the funds to me to pass on to the church in
Thailand.  The USA church says I will not be tax liable and they won't
be issuing a 1099 to me.  What do I need to do, if anything, to avoid
USA tax liability for helping out this way?

Nice, that you want to help.

This is just a free comment.  Also note from the disclaimer below that
it is no legal or professional advice.

As long as you can document that you passed the funds on to the Thai
organization, it should be no problem.  Get a receipt, if you pay it
in cash, and ask for a copy of a letter from the organization to the
US church confirming receipt of the amount and that it was transferred
via you.  The US church will want this anyway.

Enjoy Thailand.