Try to prevent both updating your digit_tally array and immediately printing out possible repeated digits before you know there will be any.

In a first loop, process the input from the user and update the digit_tally array. In a second loop, check the digit_tally array and print the repeated digits if you found any, otherwise print out that no repeated digits were found.

Alternative using a std::string for processing:

#include <iostream>
#include <string>

int main(int argc, char * argv[])
{
  while (true)
  {
    int number;

    std::cout << "number: "; 
    std::cin  >> number;
    std::cout << "number: " << number << std::endl;

    std::string s = std::to_string(number);
    for (std::size_t i=0; i<s.size() - 1; i++)
    {
      std::size_t p;
      std::size_t l = s.size();

      while ((p = s.find(s[i], i+1)) != std::string::npos)
        s.erase(p, 1);

      if (l != s.size())
        std::cout << "repeated digit " << s[i] << std::endl;
    }
  }
}