Zero the byte (eventually)

Python 2, 39 bytes

f=lambda a,d,p=1:p%256and-~f(a+d,d,p*a)

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Tracks the product p of the arithmetic progression until it's divisible by 256.

APL (Dyalog), 24 bytes


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is the the sequence (starting with the first item), the step.

0=256|×/⍵ - if the product is divisible by 256, ≢⍵ return the length.

⍺∇⍵ - else, recurse, ,⍨⊃⍺+⍵ - and append a new term to the sequence.

Raku, 38 31 bytes


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Does pretty much exactly what the description asks for. Generates the arithmetic sequence until the product of elements is divisible by 256 and returns the length of the list.