OFDM sensitivity to sampling time offset (with half of time interval between samples)
Yes, fine timing synchronization is still very important for OFDM. As you know, an FFT is typically used to demodulate an OFDM signal once it's been properly synchronized and stripped of cyclic prefix. To see what would happen if you aren't finely synchronized in time, recall the time-shifting property of the DFT:

$$
x[n-l] \Leftrightarrow X[k] e^{j\frac{2\pi kl}{N}}
$$

That is, an arbitrary timing offset $l$ (which need not be an integer for this formula to be valid) causes a linear phase shift in each output bin that is proportional to its distance from zero frequency. 

So, your subcarriers near the center of the band will not be affected that much, but as you move outward, your demodulated signals will experience an increasing amount of phase rotation. That phase rotation is related to the unknown timing error $l$, and this value will likely vary slightly from symbol period to symbol period (assuming you don't have perfect symbol rate synchronization either). This all adds up to degraded performance.

The solution? You need fine symbol synchronization in OFDM, just like with any other digital modulation format.