To probe manifestations of multiband superconductivity in oxypnictides, we measured the angular dependence of the magnetic torque $tau(theta)$ in the mixed state of LaO$_{0.9}$F$_{0.1}$FeAs single crystals as a function of temperature $T$ and magnetic fields $H$ up to 18 T. The paramagnetic contribution of the Fe ions is properly treated in order to extract the effective mass anisotropy parameter $gamma=(m_c/m_{ab})^{1/2}$ from $tau(theta)$. We show that $gamma$ depends strongly on both $T$ and $H$, reaching a maximum value of $sim$ 10 followed by a decrease towards values close to 1 as $T$ is lowered. The observed field dependencies of the London penetration depth $lambda_{ab}$ and $gamma$ suggest the onset of suppression of a superconducing gap at $H approx H_{c2}/3$.