How to programmatically specify multiple iterators?

What you actually want is to create a Sequence from the Table to be used as your iterators.

You can do this with

Do[Print[B[0] + B[1] + B[2]], Sequence @@ Table[{B[i], 0, i}, {i, 0, 2}] // 
Evaluate]

(*0
  1
  2
  1
  2
  3*)

Or, so you don't have to force evaluation,

Do[Print[B[0] + B[1] + B[2]], ##] & @@ Table[{B[i], 0, i}, {i, 0, 2}]

Perhaps you can avoid Do and instead use Tuples:

Tuples @ Range[0, {0, 1, 2}]

{{0, 0, 0}, {0, 0, 1}, {0, 0, 2}, {0, 1, 0}, {0, 1, 1}, {0, 1, 2}}

You can then use Total to sum each tuple:

Total[
    Tuples @ Range[0, {0, 1, 2}],
    {2}
]

{0, 1, 2, 1, 2, 3}

Do[Print[B[0] + B[1] + B[2]], 
 Evaluate[Sequence @@ First@ Row[Table[{B[i], 0, i}, {i, 0, 2}], ","]]]

row = Row[Table[{B[i], 0, i}, {i, 0, 2}], ","];
Do[Print[B[0] + B[1] + B[2]], Evaluate[Sequence @@ row[[1]]]]

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