Answer
The acceleration is less than $\frac{F}{m}$
Work Step by Step
We can write an expression for the relativistic mass of an object when it is moving at speed $v$:
$m_{rel} = \frac{m}{\sqrt{1-\frac{v^2}{c^2}}}$
We can find an expression for the acceleration:
$F = m_{rel}~a$
$a = \frac{F}{m_{rel}}$
$a = \frac{F}{\frac{m}{\sqrt{1-\frac{v^2}{c^2}}}}$
$a = \frac{F~\sqrt{1-\frac{v^2}{c^2}}}{m} \lt \frac{F}{m}$
The acceleration is less than $\frac{F}{m}$.
