Entanglement, measurement, and classical communication together enable teleportation of quantum states between distant parties, in principle with perfect fidelity.  To what extent do correlations and entanglement of a many-body wavefunction transfer under imperfect teleportation protocols?  We address this question for the case of teleporting the ground state of the Ising chain, focusing on its quantum critical wavefunction.  We demonstrate that imperfections, e.g., in the entangling gate adopted for a given protocol, effectively manifest as weak measurements acting on the otherwise pristinely teleported critical state.  Armed with this perspective, we leverage and further develop the theory of measurement-altered quantum criticality to quantify the resilience of critical-state teleportation. We identify classes of teleportation protocols for which imperfection (i) preserves both the universal long-range entanglement and correlations of the original quantum critical state, (ii) weakly modifies these quantities away from their universal values, and (iii) obliterates long-range entanglement altogether while preserving power-law correlations, albeit with a new set of exponents.  We also show that mixed states describing the average over a series of sequential imperfect teleportation events retain pristine power-law correlations due to a `built-in' decoding algorithm, though their entanglement structure measured by the negativity depends on errors similarly to individual protocol runs.  These results may allow one to design teleportation protocols that optimize against errors---highlighting a potential practical application of measurement-altered criticality.