How do I best support my boyfriend with the loss of his father?

Tremendously close? What do you mean? How close? Is you boyfriend dependent on his Father still, and how old it your boyfriend? Is he acting “tremendously” sad?

William

Well, I'd say, use your own experiences -- certainly you have an understanding of what he might not want to hear (some people are comforted by expressions like "God called him back" whereas others are repulsed by that and it hurts them) and little acts that you can do which might be quietly comforting (e. g. if you run the errands for one week, food and clean laundry magically appear, sometimes that is a huge relief for people -- even though they could have done it themselves you are still helping them out in a big way).

But also recognize that your experiences are not universal -- no one's are -- so what works for you may not work for him, no matter how well you know and understand him. So I would say -- talk. He may or may not want to talk about things now. But let him know that you are available when he does want to express himself.

Just be there. Help out the family with errands, food, thank you notes, transportation, etc.
They are all numb from shock and it will take a while for it all to sink in.
He will need you later.
Just tell him you are there and if there's anything he needs . . .
Find out about grief groups in the area that he/they can attend in about 3 months.
Good luck, but get some professional help resources for the family if he or any other member shows that they need extra guidance.