Design an efficient algorithm to sort 5 distinct keys in fewer than 8 comparisons

Compare A to B and C to D. WLOG, suppose A>B and C>D. Compare A to C. WLOG, suppose A>C. Sort E into A-C-D. This can be done with two comparisons. Sort B into {E,C,D}. This can be done with two comparisons, for a total of seven.

This is pseudocode based on Beta's answer. Might have some mistakes as I did this in a hurry.

if (A > B)
    swap A, B
if (C > D)
    swap C, D
if (A > C)
    swap A, C
    swap B, D  # Thanks Deqing!

if (E > C)
    if (E > D)  # A C D E
        if (B > D)
            if (B > E)
                return (A, C, D, E, B)
            else
                return (A, C, D, B, E)
         else
            if (B < C)
                return (A, B, C, D, E)
            else
                return (A, C, B, D, E)

    else  # A C E D
        if (B > E)
            if (B > D)
                return (A, C, E, D, B)
            else
                return (A, C, E, B, D)
         else
            if (B < C)
                return (A, B, C, E, D)
            else
                return (A, C, B, E, D)
else
    if (E < A)  # E A C D
        if (B > C)
            if (B > D)
                return (E, A, C, D, B)
            else
                return (E, A, C, B, D)
         else
             return (E, A, B, C, D)

    else  # A E C D
        if (B > C)
            if (B > D)
                return (A, E, C, D, B)
            else
                return (A, E, C, B, D)
         else
            if (B < E)
                return (A, B, E, C, D)
            else
                return (A, E, B, C, D)

It has to be 7 or more comparisons.

There are 120 (5 factorial) ways for 5 objects to be arranged. An algorithm using 6 comparisons can only tell apart 2^6 = 64 different initial arrangements, so algorithms using 6 or less comparisons cannot sort all possible inputs.

There may be a way to sort using only 7 comparisons. If you only want to sort 5 elements, such an algorithm could be found (or proved not to exist) by brute force.