Table Lookup

def roman(number):
    assert (number > 0)

    # define lookup table (as a tuple of tuples, in this case)
    table = (
        ("I", "II", "III", "IV", "V", "VI", "VII", "VIII", "IX"),
        ("X", "XX", "XXX", "XL", "L", "LX", "LXX", "LXXX", "XC"),
        ("C", "CC", "CCC", "CD", "D", "DC", "DCC", "DCCC", "CM"),
        ("M", "MM", "MMM"))

    # convert the input integer to a list of single digits
    digits = [int(d) for d in str(number)]
    
    # we need the row in the lookup table for our most-significant decimal digit
    inverter = len(digits) - 1 

    # translate decimal digits list to Roman numerals list
    roman_digits = [table[inverter - i][d - 1] for (i, d) in enumerate(digits) if d != 0]

    # convert the list of Roman numerals to a single string
    return ''.join(roman_digits)

In this approach we loop over decimal digits, not their Roman equivalents.

The key point is to have a 2-dimensional lookup table, with each row corresponding to a separate digit: ones, tens, hundreds, thousands. Each digit can then be converted to its Roman equivalent with a single lookup.

Note that we need to compensate for Python's zero-based indexing by (in effect) subtracting 1 from each row and column.

Optional modification

In the code above, we used the inverter variable to work bottom-to-top through the lookup table. This allows working left-to-right through the decimal digits.

Alternatively, we could reverse the digits list, go top-to-bottom through the lookup table, then reverse the roman_digits list before the final join().

def roman(number):
    assert (number > 0)

    # define lookup table (as a tuple of tuples, in this case)
    table = (
        ("I", "II", "III", "IV", "V", "VI", "VII", "VIII", "IX"),
        ("X", "XX", "XXX", "XL", "L", "LX", "LXX", "LXXX", "XC"),
        ("C", "CC", "CCC", "CD", "D", "DC", "DCC", "DCCC", "CM"),
        ("M", "MM", "MMM"))

    # convert the input integer to a list of single digits, in reverse order
    digits = [int(d) for d in str(number)][::-1]

    # translate decimal digits list to Roman numerals list
    roman_digits = [table[i][d - 1] for (i, d) in enumerate(digits) if d != 0]

    # reverse the list of Roman numerals and convert to a single string
    return ''.join(roman_digits[::-1])

This eliminates one line of code, at the cost of adding two list reverses.

The [::-1] indexing is idiomatic Python, but less experienced programmers may not find it very readable.

Credit

This approach was adapted from one created by @cmcaine on the Julia track.