Excludivisible numbers

Jelly, 21 19 bytes

DLR©œ^€®ịDḌḍ@DPȧµ#Ṫ

Input is 1-indexed. Try it online!

How it works

DLR©œ^€®ịDḌḍ@DPȧµ#Ṫ  Main link. No arguments.

                µ    Convert the chain to the left into a link.
                 #   Read n from STDIN and execute the link to the left for
                     k = 0, 1, 2, ... until n value return a truthy value.
D                      Convert D to base 10 (digit array).
 LR                    Construct the range from 1 to the length of the digit array.
   ©                   Copy the range to the register.
       ®               Yield the value of the register.
    œ^€                Take the multiset difference of each element of the range
                       and the range itself.
        ịD             Index into the digit array.
          Ḍ            Convert each list of digits to integer.
                       This yields all numbers with one suppressed digit.
           ḍ@D         Test each of these numbers for divisibility by the
                       suppressed digit from the digit array.
              P        Take the product of the resulting Booleans.
               ȧ       Logical AND with k (k would return 1).
                  Ṫ  Tail; extract the last (nth) element.

Pyth, 20

e.f!s.e.x%s.D`Zksb1`

Try it here or run the Test Suite

N is 1-indexed for this answer.

Uses largely the same logic as my python script. Notable differences are using .D to delete the digit from the string during the test and using exception handling to deal with zero digits.

Pyth, 26 bytes

e.f&!/J`Z\0!s%VsM.DLJUJjZT

Test suite.

e.f&!/J`Z\0!s%VsM.DLJUJjZT
e.f                          The n-th number (Z) where:
    !/J`Z\0                      Z does not contain "0"
   &                             and
           !s                    the following does not contain zero:
               sM.DLJUJ              generate the set of the digits removed
                       jZT           all the digits
             %V                      modulus in parallel
                                     if Z is 324,
                                     the first set would be [24,34,32]
                                     the second set would be [3,2,4]
                                     the third set would be [24%3, 34%2, 32%4]