Mutation That Loses Stop Codon
No, this will not happen. mRNAs are inspected in the nucleus before they are exported into the cytoplasm (at least in eukaryotes), where transcription and translation don't happen at the same place. This ensures that no mRNAs without stop codons or premature stop codons are exported. This phenomenon is called
"mRNA surveillance". mRNAs that do not pass this quality check, are degraded. See the Wiki reference for some basic information and the references below for more.

References:


Process or perish: quality control in mRNA biogenesis.
The exosome and RNA quality control in the nucleus
A faux 3′-UTR promotes aberrant termination and triggers
nonsense-mediated mRNA decay