First, it's important to understand what constitutes "finishing" a race in Track and Field. The winner is not the first person to get a body part across the line or "touch the tape" as you referred. The winner is the first person to get their torso across the finish line.
A situation similar to this occurred in the US Olympic trials for the 2012 London Olympics. Allyson Felix and Jeneba Tarmoh tied for 3rd (which was important because only the top 3 qualify for the Olympic team). Originally, Tarmoh was declared the winner by .0001 seconds, but later they determined it was a tie (the cameras they used shot at 3000 fps - quite capable of determining a tie).
If this were to happen in the Olympics, both would be awarded the bronze medal. If the tie were for 1st, they would both be awarded a gold medal and the 3rd finisher would be awarded a bronze and no silver would be awarded (This happened in 2000 when Gary Hall Jr. and Anthony Ervin tied in the 50M freestyle swim).
However, because only 3 members of team could qualify, a tie breaker was necessary. The USATF (nor the IAAF - the international governing body of T&F) had no rules in place to settle this particular tie, so they resorted to the following:
- There will either be a coin flip or a run off.
- Each runner will be allowed to pick which option they want. If they disagree, the tiebreaker will be a runoff.
- If both runners decide not to pick an option, the tiebreaker will be a coin flip.
Eventually Jenebah Tarmoh pulled out from the tie-break and surrendered the spot to Felix.
A situation like this is obviously rare and will probably only happen in shorter distances.