Are both doors on the same orientation on the house (example they both face east?).

Wind patterns will have some effect. As well as infiltration and air currents related to how tight your house envelope is and where are comes in and leaves.

But basically heat travels from warmer to cooler by the path of least resistance. If the overall opening is significantly larger on one door or the other that might make some difference.. If the larger opening door would be open for a shorter duration, that may cancel out the size differential.

But how it makes your house feel will be the true test for what is best for you to do.

You really asked 2 different questions/statements. Are you interested in the theoretical of the heat loss .....or how warm the great room stays? Not opening the door to the outside in the great room I would suspect would keep that room warmer.

Use the mudroom door - it's an airlock.

I didn't think of that Lazy . If the mud room door does not open directly outside, or if there is a second door between the mud room and main living space, then most definately that option is more efficient.

thanks! both face west, and the mud room is tiny--8x12--and overflowing at this time of year.
the kitchen is between the mud room and great room, which i despise, as it funnels every visitor through the kitchen into the house, and then creates a huge bottleneck since no one ever wants to leave the kitchen!
every, and i mean every, gathering we have consists of people crammed into the aisle of the ktichen and not one person in the great room. not one! grr...
the kitchen is kept warm with the vintage vulcan range which has approx one thousand pilot lights going, so none of the cold air reaches the great room.

where the slider is should be the formal entry,it is larger and more centrally located, and we could then appropriate the mud room space for the kitchen which would also make sense.no traffic through but more counter and storage would be nice.
my dh is aghast at the thought of replacing the slider with a wall and standard entry door, however, so i am thinking of just using the slider anyway on occasion.

especially tuesday, as a realtor is coming to give me an idea of fair market value, and i want to show off the house as best i can, which i think means using that door even if just for that visit.
because the entry is the first impression and the last, i hope entering the house this way will make a better impression.

As the primary entrance door, I'd go with swinging french doors rather than the slider. It's probably a wash energy-wise. If I had to guess, I'd say the slider is better as you wouldn't have to open it as far and the theoretical time it's open would be less (no having to clear the swinging door).

I get you on the entry into the kitchen. We have a door entry to our garage off the front portico and then a door into the house from the garage (no mudroom). Everyone, except formal guests, enters through that stupid door and comes directly into my miniature kitchen. And they stay there. I have a beautiful formal living room that you'd enter from the front door, but no one ever comes in there except when we're having a party with infrequent visitors.

I want to close off the entry door but DH uses it as the door for his customers, so no go, and the door from the kitchen leads out to my laundry room in the garage. I'm stuck!