A friend of mine, Adam, has his father pay his credit card bills, as
part of his compensation for working for him part time. He also owns
his own business. Can he take tax deductions on the business expenses
he incurs on that card?

I believe that the rules state that you must have actually paid for
something if you are going to take a deduction for it.  It seems like
this arrangement of the father paying the son by paying for a credit
card is likely to avoid payroll taxes that would be due should the
father pay the son by a regular paycheck.  As they say, you can't have
your cake and eat it too.  This person is probably cheating the IRS by
having this payment arrangement, so I don't think you'd be able to use
the arrangement to avoid even more taxes.  Eventually this will catch
up to them.