The properties of a phase at finite interactions can be significantly influenced by the underlying dispersion of the noninteracting Hamiltonian. We demonstrate this by studying the repulsive Hubbard model on the two-dimensional Lieb lattice, which has a flat band for vanishing interaction U. We perform real-space dynamical mean-field theory calculations at different temperatures and dopings using a continuous-time quantum Monte Carlo impurity solver. Studying the frequency dependence of the self-energy, we find that a nonmagnetic metallic region at finite temperature displays non-Fermi-liquid behavior, which is a concomitant of the flat-band singularity. At half-filling, we also find a magnetically ordered region, where the order parameter varies linearly with the interaction strength, and a strongly correlated Mott insulating phase. The double occupancy decreases sharply for small U, highlighting the flat-band contribution. Away from half-filling, we observe the stripe order, i.e., an inhomogeneous spin and charge density wave of finite wavelength, which turns into a sublattice ordering at higher temperatures.