I put some small 1/2" holes around the trunk on a river birch tree and gave it an iron treatment. The instructions called to seal the holes with latex caulk, but I only have latex caulk with silicone added in. Is it a problem to use it with silicone to seal small holes in the tree?
There is no need to seal the holes at all. The natural healing process that woody plants use is to grow around the wound and eventually cover it over.
Take a break and let nature do the job for you...
It's not a problem. The tree will protect itself from the wounds, which by your description seem fairly minor (unless it is a small, immature tree). Room temperature vulcanizing silicone releases a small amount of acetic acid while curing. While it is toxic and slightly corrosive, the small amount released by a siliconized latex caulk compared to a non-silicone latex caulk would be insignificant.
By the way, IMO you should use a pruning sealer instead.