If feeding sugar syrup in a Bee Tree Hive is what you're wondering about, please read to the end of this post.
Bee Tree Hives are designed to support bee-centric beekeeping, which puts the long-term health, welfare, and vitality of honey bees foremost. Bee-centric beekeeping does not support feeding sugar syrup to honey bees, and therefore, neither does Bee Tree Hives.
And, as a bee-centric beekeeper with the goal of developing an ecotype (a genetically distinct geographic variety, population or race within a species, which is adapted to specific environmental conditions), I have many of my own, more specific reasons for not feeding sugar syrup to honey bees.
In addition, apart from my goals as a beekeeper, the feeding of sugar syrup to honey bees is, at its core, an ethical issue for me. For me, replacing honey with sugar syrup is the equivalent of taking good food from your children or grandchildren and then feeding them Twinkies all winter... because you want their good food.
That said, however, I readily concede that if Twinkies were all the food you had to keep your children or grandchildren alive through the winter, then you would absolutely feed them Twinkies.
All of my colonies are located between 7,000' and 9,000' in elevation here in the heart of the Rocky Mountains, and our valley is classified as "high mountain desert." It has this classification because we get less than 10" of precipitation each year; it is extremely arid (the humidity is typically less than 20%); and, because the air is so clear, the sun is very intense. As you might imagine, then, our nectar-flow season is both short and lean.
So, given my heart as a beekeeper, my goals as a beekeeper, and the region of the country in which I steward honey bees, I harvest no honey for myself. Any surplus honeycomb is stored to be fed back to any colonies that might need it in an emergency.
I can imagine, however, that there might come a time when a colony, through some disaster it couldn't predict or control, doesn't end up with enough honey to make it through the winter. And, if this happened to multiple colonies, maybe I wouldn't have enough stored honeycomb to go around. And, perhaps, these bees might otherwise be well-adapted to my region. In this kind of scenario, I would feed sugar syrup if that is the only way I could hope to get the bees through the winter.
So, can sugar syrup be fed to bees in a Bee Tree Hives hive cover? Of course. The feeding chamber where I place pieces of broken comb, or whole frames of honeycomb, can also receive a plastic container as much as 12" wide and 15" long. There would just need to be floats placed on the sugar syrup so that the bees don't drown in it.
All the other benefits of feeding in a Bee Tree Hive still remain. The access door can be opened without exposing the brood nest, disturbing the bees, or even breaking any of their propolis seals. In the winter, you are briefly letting a little bit of heat out, so you want to be quick, but that is a worthwhile trade-off if the bees need to be fed.
P.S., feeding bees inside the hive, even in the summer, is a topic for another post. There are many reasons why this is in the best interest of the bees. (And I've learned most of them the hard way.)