Again, sort of...
A 4runner doesn't need anywhere near a 116 rated tire to carry the load required. It's nice that it comes stock but is unnecessary based on the weight/load capacity of the 4runner. Accordingly, when you run a C rated tire on a 4runner you only need to support the load required. Doesn't mean you need to match the 116 rating of the oem tire. For example a Caddy Escalade ESV comes stock with 111 rated tires and its a heavy beast.
As far as tire pressure goes: https://www.tirerack.com/tires/tiret...jsp?techid=196
So, temp and pressure rise are expected. Again match your load to your tire pressure and nothing to worry about.