How precisely can I track distances using the iPhone accelerometer?

Theoretically the accelerometers in the iPhone can be used to track distance using interpolation over time.

What is the estimated error for tracking distances between 1 and 10 meters?
Is there any iPhone app that work as a measuring tape?

From what I've heard, the iPhone accelerometer output is very noisy - most applications that utilize it for any sort of moderate precision work do a lot of filtering and output modification to smooth out the noise. The iPhone can be placed flat on a table, and the accelerometer will still output a fair bit of data. I imagine you could certainly use it to measure longer distances (>10m), but I probably wouldn't trust it for high resolution measurements.

The accelerometer chips used in the iphone have too much noise to be used for accurate distance integration over even short distances with simple movements. Even if it were a quiet sensor and provided good data, Apple pre-filters the accelerometer in a way that prevents applications from performing a proper integration. The filtering is useful to lower the terrible noise floor, and to make app developer's lives easier whether they want 1 update per second or 100, but it's irreversible and Apple doesn't provide a way to get the raw data, so any acceleration integration only produces delayed distance with significant overshoot. This is, of course, on top of the normal problems of using an accelerometer to find distance (removing gravity - which the gyroscope makes much easier - for instance).

Most of the pedometer apps use the accelerometer to count steps and use the GPS to measure distance over longer distances and to calibrate the user's individual step distance. They do not sense distance using the accelerometer.