Can tight hamstrings ruin my posture?
It can cause problems indirectly. Tight hamstrings will pull the pelvis into a less than ideal position. This in turn will put strain on the lower back, which can affect your posture.
Tight hamstrings also means you are not as flexible. If you run, your stride is not at its optimal length. Thus, you end up taking a few more steps to cover the same distance. It also increases the odds of you getting injured if you have to make a sudden change in speed.