Using SageMath I found that the divisibility graph of the set $\{2021,2022,2023,\dots,N\}$ is planar for all $N$ up to and including
$15119$.
That it is not planar when
$N=15120$
follows from the fact that now the graph includes a subdivision of $K_{3,3}$, forbidden by Kuratowski's theorem for planar graphs:

SageMath also shows that for the set $\{2022,2023,\dots,N\}$, the value of
$15119$
remains as the most $N$ can be for the graph to be planar beginning with 2022 and beyond, maybe as far as 2047 (due to the presence of 2048 in the graph above).
The $N$th term of the sequence (likewise found with the help of SageMath) $14,23, 29,\dots,783$ gives the maximum value of $N$ such that $\{1,2,3,\dots,N\}$ is planar.
The first hundred terms of the sequence are as follows:
14, 23, 29, 39, 44, 47, 69, 79, 83, 89, 95, 95, 119, 119, 119, 143, 143, 143, 167, 167, 188, 191, 191, 191, 224, 224, 224, 224, 224, 224, 269, 269, 279, 279, 279, 279, 335, 335, 335, 335, 335, 335, 359, 359, 359, 383, 383, 383, 419, 419, 431, 431, 431, 431, 449, 449, 449, 449, 449, 449, 503, 503, 503, 503, 524, 524, 524, 524, 524, 524, 524, 524, 599, 599, 599, 623, 623, 623, 671, 671, 671, 671, 671, 671, 674, 674, 674, 674, 674, 674, 767, 767, 767, 767, 767, 767, 783, 783, 783, 783.