#JavaScript (ES6),  62 60  59 bytes

Takes input as a 0-padded string. Returns \$0\$ for false or a positive integer for true.

s=>(m=s[2]+s[3])<13&31-(m^2?~m%9%2:s%4?1:2)>(s/=1e4)&&~~s*m

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or Check all possible outputs against an ungolfed implementation

###Commented

s =>                // s = input string
  (m = s[2] + s[3]) // m = month, as a string
  < 13 &            // make sure that m is less than 13
  31 - (            // compute the upper bound for this month:
    m ^ 2 ?         //   if the month is not February:
      ~m % 9 % 2    //     use either 31 or 32
    :               //   else:
      s % 4 ?       //     if this is not a leap year:
        1           //       use 30
      :             //     else:
        2           //       use 29
  )                 // end of upper bound computation
  > (s /= 1e4)      // make sure that it's greater than the day
  && ~~s * m        // and finally make sure that day * month is not zero

JavaScript (ES6),  62 60  59 bytes

Takes input as a 0-padded string. Returns \$0\$ for false or a positive integer for true.

s=>(m=s[2]+s[3])<13&31-(m^2?~m%9%2:s%4?1:2)>(s/=1e4)&&~~s*m

Try it online!

or Check all possible outputs against an ungolfed implementation

Commented

s =>                // s = input string
  (m = s[2] + s[3]) // m = month, as a string
  < 13 &            // make sure that m is less than 13
  31 - (            // compute the upper bound for this month:
    m ^ 2 ?         //   if the month is not February:
      ~m % 9 % 2    //     use either 31 or 32
    :               //   else:
      s % 4 ?       //     if this is not a leap year:
        1           //       use 30
      :             //     else:
        2           //       use 29
  )                 // end of upper bound computation
  > (s /= 1e4)      // make sure that it's greater than the day
  && ~~s * m        // and finally make sure that day * month is not zero

#JavaScript (ES6),  62 60  59 bytes

Takes input as a 0-padded string. Returns \$0\$ for false or a positive integer for true.

s=>(m=s[2]+s[3])<13&31-(m^2?~m%9%2:s%4?1:2)>(s/=1e4)&&~~s*m

Try it online!

###Commented

s =>                // s = input string
  (m = s[2] + s[3]) // m = month, as a string
  < 13 &            // make sure that m is less than 13
  31 - (            // compute the upper bound for this month:
    m ^ 2 ?         //   if the month is not February:
      ~m % 9 % 2    //     use either 31 or 32
    :               //   else:
      s % 4 ?       //     if this is not a leap year:
        1           //       use 30
      :             //     else:
        2           //       use 29
  )                 // end of upper bound computation
  > (s /= 1e4)      // make sure that it's greater than the day
  && ~~s * m        // and finally make sure that day * month is not zero

#JavaScript (ES6),  62 60  59 bytes

Takes input as a 0-padded string. Returns \$0\$ for false or a positive integer for true.

s=>(m=s[2]+s[3])<13&31-(m^2?~m%9%2:s%4?1:2)>(s/=1e4)&&~~s*m

Try it online!

or Check all possible outputs against an ungolfed implementation

###Commented

s =>                // s = input string
  (m = s[2] + s[3]) // m = month, as a string
  < 13 &            // make sure that m is less than 13
  31 - (            // compute the upper bound for this month:
    m ^ 2 ?         //   if the month is not February:
      ~m % 9 % 2    //     use either 31 or 32
    :               //   else:
      s % 4 ?       //     if this is not a leap year:
        1           //       use 30
      :             //     else:
        2           //       use 29
  )                 // end of upper bound computation
  > (s /= 1e4)      // make sure that it's greater than the day
  && ~~s * m        // and finally make sure that day * month is not zero

#JavaScript (ES6),  62 60  59 bytes

Takes input as a 0-padded string. Returns \$0\$ for false or a positive integer for true.

s=>(m=s[2]+s[3])<13&31-(m^2?~m%9%2:s%4?1:2)>(s/=1e4)&&~~s*m

Try it online!

###Commented

s =>                // s = input string
  (m = s[2] + s[3]) // m = month, as a string
  < 13 &            // make sure that m is less than 13
  31 - (            // compute the upper bound for this month:
    m ^ 2 ?         //   if the month is not February:
      ~m % 9 % 2    //     use either 31 or 32
    :               //   else:
      s % 4 ?       //     if this is not a leap year:
        1           //       use 30
      :             //     else:
        2           //       use 29
  )                 // end of upper bound computation
  > (s /= 1e4)      // make sure the upper bound is greater than the day
  && ~~s * m        // and finally make sure that day * month is not zero

#JavaScript (ES6),  62 60  59 bytes

Takes input as a 0-padded string. Returns \$0\$ for false or a positive integer for true.

s=>(m=s[2]+s[3])<13&31-(m^2?~m%9%2:s%4?1:2)>(s/=1e4)&&~~s*m

Try it online!

###Commented

s =>                // s = input string
  (m = s[2] + s[3]) // m = month, as a string
  < 13 &            // make sure that m is less than 13
  31 - (            // compute the upper bound for this month:
    m ^ 2 ?         //   if the month is not February:
      ~m % 9 % 2    //     use either 31 or 32
    :               //   else:
      s % 4 ?       //     if this is not a leap year:
        1           //       use 30
      :             //     else:
        2           //       use 29
  )                 // end of upper bound computation
  > (s /= 1e4)      // make sure that it's greater than the day
  && ~~s * m        // and finally make sure that day * month is not zero