Is the temperature (even a bit) warmer in a forest?
It will depend on the time of year and the time of day. As you noted, you often see more frost in an open field than in a forest. A comment indicated the forest insulated the surface but I believe it will also reduce radiant heat loss at night, compared to the open field. During the day, the ground surface is more shaded and the trees reduce the advective movement of warmer air into the forest floor so the forest will often be cooler than the open field.

Another factor is evapotranspiration. Forests will tend to be cooler in drier summer conditions because, in general, you get more evapotranspiration from trees than from shallow-rooted vegetation and annual grasses die off in summer. The evapotranspiration cools the area.