If you are sleeping in a rainstorm it would be warmer to sleep under a tarp, since otherwise you are going to get wet.
It would be rather difficult to enclose a space with a tarp to the level that a tent provides and so I am wondering if setting up a tarp and sleeping underneath it on a calm night would provide more warmth compared to sleeping out under the stars. Wind should be neglected here.
I have spent many nights sleeping in tents, tarps, and under the stars.
Tents are definitely your warmest option, and can be up to 10 degrees warmer than the outside - if you're using a rainfly. Just using the mesh tent body will even increase the temperature by a few degrees, but won't protect from any weather except mild wind.
Tarps can certainly increase the temperature too, but as @Wills mentioned there are some other variables to take into account. If you attach your tarp low to the ground (reducing air volume and also cross-breeze) you can trap some warmth around you. Planning your placement so that your tarp does not create a wind tunnel, and instead catches breezes broadside, will increase your ability to trap heat. Further reducing air flow by surrounding natural elements or using your pack/equipment to block the doorway works too. In many cases, the disadvantage with temperature is far outweighed by the decrease in weight offered by the tarp. edit: compared to a tent, not to sleeping under the stars.
The shape of the land affects sleeping temperature too - ex: cold air will settle in dips in the ground. Water (streams and lakes) also have a chilling effect on air temperature.
Don't forget that most heat loss while camping actually occurs through the ground (given the calm night you mention). A quality sleeping pad will go a long way in keeping you warm.
A tarp is significantly warmer.
On a clear night the surface of your bag is radiating into interstellar space. Not much radiation back. With a tarp, you have one absorb/reradiate layer which will essentially cut this in half. If you look on a frosty morning, you will see that there is often no frost under a tree. Due to the tree intercepting radiation from the ground. Ground didn't cool as fast.
Due to less radiation the surface of your bag doesn't get cold enough to collect nearly as much frost or dew. So your second night will be warmer.
If there is a wind, then the tarp is effective is so far as it slows down the wind.
If you pitch your tarp too low, you will have condensation where your sleeping bag touches the tarp.
I have found generally that I will stay drier with a tarp than I will in a tent. Most of my camping has been in either the Canadian Rockies, or the Pre-Cambrian Shield. Below timberline, not in established camps. My companions had to make compromises to find locations for their tents. This resulted in rain getting into the tent. With a tarp, I had a much easier time setting up in a spot that would remain dry, plus the advantage of a dirt/grass/moss/gravel floor meant that water that did get in was more likely to sink than to soak. (I carry a groundsheet too, but it has a much smaller footprint than the tarp.
This is much less true if it's snowing.
Neglecting the wind (advection) the tarp shouldn't make any noticeable difference. I doubt you could measure even one Kelvin difference.
Even a tent isn't providing a lot of warmth during night but as long as you stay active in the tent (by moving and/or cooking) you will notice an increase in the temperature. During night you are passive and most of the produced body heat should be restrained in the sleeping bag if you use it right (warm enough bag, as little openings as possible). If you (with you I mean the system of you and your sleeping bag!) release too much heat, this is bad for your comfort level, so there is not that much energy to heat up the volume of air in the tent. That being said, the smaller that volume of trapped air in the tent, and the less ventilation the tent has, the more will the air be heated up.
In a tarp setup the ventilation and the "volume of air" is extremely high, mathematically spoken: tends infinity. Even surrounded by dense vegetation a tarp won't increase the temperature underneath noticeably.
Another source of heat, or better said a source of heat loss is heat radiation. During night - especially clear nights - the heat radiation cools the atmosphere quite a lot. If you'd have a very big tarp, this source of heat loss should be reduced, therefore helping you to heat up. But for a normal sized tarp I doubt you feel the difference.
