We study the dynamics and readout of topological qubits encoded by zero-energy Majorana bound states in a topological superconductor. We take into account bosonic modes due to the electromagnetic environment which couple the Majorana manifold to above-gap continuum quasiparticles. This coupling causes the degenerate ground state of the topological superconductor to be dressed in a polaronlike manner by quasiparticle states and bosons, and the system to become gapless. Topological protection and hence full coherence is only maintained if the qubit is operated and read out within the low-energy spectrum of the dressed states. We discuss reduction of fidelity and/or visibility if this condition is violated by a quantum-dot readout that couples to the bare (undressed) Majorana modes. For a projective measurement of the bare Majorana basis, we formulate a Bloch-Redfield approach that is valid for weak Majorana-environment coupling and takes into account constraints imposed by fermion-number-parity conservation. Within the Markovian approximation, our results essentially confirm earlier theories of finite-temperature decoherence based on Fermi’s golden rule. However, the full non-Markovian dynamics reveals, in addition, the fidelity reduction by a projective measurement. Using a spinless nanowire model with p-wave pairing, we provide quantitative results characterizing these effects.