Charcoal, 17 bytes

¬№﹪⌕Ａ⭆Ｓ∨﹪Ｉι⁷χ0⁷¦⁵

Try it online! Link is to verbose version of code. Outputs a Charcoal boolean i.e. - for haiku, nothing if not. Works like my Retina answer, except it replaces 0 and 7 with 10 and checks that neither the 6th nor the 13th syllable is 0. Explanation:

      Ｓ             Input string
     ⭆              Map over characters and join
          ι         Current character
         Ｉ          Cast to integer
        ﹪  ⁷        Reduce modulo 7
       ∨    χ       Replace zeros with predefined variable 10
   ⌕Ａ        0      Find all occurrences of literal string `0`
  ﹪           ⁷     Vectorised modulo 7
¬№              ⁵   Check that there are no 5s.

Charcoal, 25 bytes

≔⭆Ｓ∨﹪Ｉι⁷χθ›⁼¹⁷Ｌθ№﹪⌕Ａθ0⁷¦⁵

Try it online! Link is to verbose version of code. Outputs a Charcoal boolean i.e. - for haiku, nothing if not. Works like my Retina answer, except it replaces 0 and 7 with 10 and checks that neither the 6th nor the 13th syllable is 0. Explanation:

≔⭆Ｓ∨﹪Ｉι⁷χθ

Reduce all of the digits modulo 7, then change all 0s to 10, so that 0 represents a second syllable.

›⁼¹⁷Ｌθ

Check that there are 17 syllables, but not that...

№﹪⌕Ａθ0⁷¦⁵

any second syllables are at position equivalent to 5 (modulo 7).