My aunt is my father's guardian and has placed him in a assisted living facility. Can I take him out?

Is she POA or is she his Guardian, recognized by the Court?
If your father is cognizant ( in your profile you state he is 42) he needs neither a Guardian nor would his POA be active.
What medical condition does he have other than "depression, diabetes and mobility problems" none of these would necessarily mean that he needs a Guardian nor POA.
If she legally is his Guardian and has been granted Guardianship by the Court were you legally and properly notified of the Court Case? Did you go to Court and state that you wanted Guardianship?
You can try to obtain Guardianship but you will have to go to Court.

If your dad is not cognizant he might not be able to "agree" to you over riding the current Guardianship.

Your best chance of this would be to hire an Elder Care attorney (even though your dad is not "elder" if the age you have given is correct)
This is not easy, this is not inexpensive. And you may have to prove why you would be a better Guardian than the one chosen.

How do you mean guardian. Having POA does not make her his guardian. She had to file it with the courts and a Judge assign her. And thats after she has proved that Dad is not competent to make his own decisions. If this is what she did, then were you involved in the proceedings? If so, you could have contested her becoming guardian.

If she was assigned his guardian by a judge, then there is nothing you can do. Maybe he needed more care she was not willing to give so she placed him in an AL. Maybe her kids are paying rent to offset the cost of his AL.

If all she has is a POA and Dad is mentally competent to make decisions, he can assign u POA which revokes hers. You then can evict her children from Dads home (may take a while) and move him back. But be aware how much care he will need. Because you will be giving up a portion if not all of ur life for him.

He wants you be to his caregiver - until he doesn't. This kind of issue is very common with elderly people. They don't like where they are, so they latch onto someone and try to get their support for the idea of moving here, there, everywhere.

I'm assuming you've never been a caregiver before. Do you understand that it takes over your life? That as the patient gets sicker and sicker you must provide everything they need all by yourself? He's in assisted living now, so he clearly needs help. Do you know what they do for him there - like change his diapers, shower him, regulate and administer his meds? Do you think he will pay you to do these things? Would you have to give up a job with benefits and retirement plan to become his caregiver, where you must make sure he's safe 24/7? With no days off?

You have a lot to think about. Getting guardianship isn't free; you'll need a lawyer, have to pay court costs, and prove that you're able to do what he needs.

If your aunt is your dad's legal guardian, this would mean he is a ward of the state. You will need to apply to the court to change the guardianship. Your aunt can't "hand it over" the process will require a judge to make the decision to change the guardianship.