Hello Leo,

River birch will bleed sap quite a lot if pruned now. It is unsightly and can attract insects. It does not actually harm the tree though. 
The best time to prune is late summer through winter. Dead wood can be pruned any time, and remove anything that poses a hazard to people or property as soon as possible.

River birch is the most resistant species to bronze birch borer.

Research has shown that trees enclose wounds best when left bare. Wound sealers or paints can slow wound closure. So do not paint or seal cuts or wounds with anything. ( the exception is sealing cuts on oak trees during the growing season)

Here are some references to keep handy-

https://hort.extension.wisc.edu/articles/river-birch-betula-nigra/

https://www.purdue.edu/fnr/extension/tree-wounds-and-healing/

This manual illustrates proper pruning techniques, and has a guide to tree care, should you need it:

https://trees.umn.edu/sites/trees.umn.edu/files/files/tree_owners_manual_0.pdf

You are welcome. Glad to help.