The tiny url people like to use random tokens because then you can't just troll the tiny url links. "Where does #2 go?" "Oh, cool!" "Where does #3 go?" "Even cooler!" You can type in random characters but it's unlikely you'll hit a valid value.
Since the key is rather sparse (4 values each having 36* possibilities gives you 1,679,616 unique values, 5 gives you 60,466,176) the chance of collisions is small (indeed, it's a desired part of the design) and a good SQL index will make the lookup be trivial (indeed, it's the primary lookup for the url so they optimize around it).
If you really want to avoid the lookup and just unse auto-increment you can create a function that turns an integer into a string of seemingly-random characters with the ability to convert back. So "1" becomes "54jcdn" and "2" becomes "pqmw21". Similar to Base64-encoding, but not using consecutive characters.
(*) I actually like using less than 36 characters -- single-cased, no vowels, and no similar characters (1, l, I). This prevents accidental swear words and also makes it easier for someone to speak the value to someone else. I even map similar charactes to each other, accepting "0" for "O". If you're entirely machine-based you could use upper and lower case and all digits for even greater possibilities.
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