The book does not say, but 25% faster than a dog sled seems reasonable.
Unfortunately, the axe beak seems to have been passed over when discussing travel times of the various modes of transportation. But, we can make some reasonable estimates.
Dogsleds are described on page 20. A dogsled weighs 300 pounds, and one sled dog (use the wolf statblock) can pull 360 pounds. So we need 2 dogs per sled for a medium sized character. Two dogs pulling a medium character and a sled will travel at the usual pace of 1 mile per hour over land (pg. 11).
Now, axe beaks are not afforded the courtesy of having their overland travel speeds just given to us. Naturally, it would seem to make sense to just compare its listed speed to the listed speed of a sled dog (wolf). We just need to be sure we won't run into any issues of our axebeak becoming over encumbered. The axe beak description on page 20 says:
An axe beak’s splayed toes allow it to run across snow, and it can carry as much weight as a mule.
Okay, so it can carry as much as a mule. Why do I have to do that math? Is Christopher Perkins the one writing the convoluted math word problems we all did in high school? Looking at the mule statblock, we see this feature:
Beast of Burden. The mule is considered to be a Large animal for the purpose of determining its carrying capacity.
Okay, so it counts as large, and it has a strength score of 14. From the rules on carrying capacity:
Your carrying capacity is your Strength score multiplied by 15. This is the weight (in pounds) that you can carry, which is high enough that most characters don't usually have to worry about it.
Larger creatures can bear more weight, whereas Tiny creatures can carry less. For each size category above Medium, double the creature's carrying capacity and the amount it can push, drag, or lift. For a Tiny creature, halve these weights.
So the axebeak can comfortably carry 420 pounds - more than enough to carry a medium humanoid and their gear, and exactly the pulling weight of two dogs pulling a sled.
This is the notable point: Two dogs can pull 720 pounds, and a sled weighs 300 pounds, so two dogs pulling a sled can pull exactly 420 pounds of character, and an axe beak can pull exactly 420 pounds of character.
Two dogs and a sled can pull exactly the same amount of character as one axebeak. Therefore, it seems entirely reasonable that just comparing their listed speeds will be an apples-to-apples comparison. The wolf has a listed speed of 40 ft, the axebeak has a listed speed of 50 ft, so we conclude that the axe beak is 25% faster than two dogs pulling a dogsled.
Page 11 does have the statement:
The fastest way to travel across the tundra is by dogsled.
What do we do with this? I don't know. Ignore it? The poor axebeak seems to have been an after thought here, so it would not surprise me if this sentence was written before axebeak was even put in. It seems clear that the writers didn't think too hard about how fast an axebeak could move through tundra, so it would make sense if this statement didn't consider axebeaks at all.
Either way, it's your adventure. You decide. Unless you aren't the DM, then just buy a dogsled. As the introduction says:
Nothing in this adventure is too sacred to tamper with and repurpose to serve your own needs.
Do they need to rest?
I need to rest. I'm exhausted from Christopher Perkins’ stablock-goose-chase of a math problem. I'd say they need to rest. If you want to be faithful to the adventure's apparent intention for the difficulty of overland travel, making axebeaks faster and making them not have to rest seems to be a little cheesy. They're already way cheaper than the dogsled setup.