If you simply specify Times New Roman in css, then no, no need to pay anything. You're not hosting a font file.
For hosting a font file, yes, sure, but in that case I recommend at least checking out alternatives such as STIX Two Text (open-source, based on 8pt TNR, better for small sizes), Liberation Serif (squarer design), Tiempos (commercial, more styles and weights), Times Eighteen (sharper, designed for headlines).
STIX Two Text for smaller text and TNR for larger sizes might be a good combination. If you're using serif fonts for both body text and headings, using optical sizes makes your layout look sharper. Look here in the section on "progressive sizing" for a demonstration.