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In the submm regime, spectral line scans and line intensity mapping (LIM) are new promising probes for the cold gas content and star formation rate of galaxies across cosmic time. However, both of these two measurements suffer from field-to-field variance. We study the effect of field-to-field variance on the predicted CO and [CII] power spectra from future LIM experiments such as CONCERTO, as well as on the line luminosity functions (LFs) and the cosmic molecular gas mass density that are currently derived from spectral line scans. We combined a 117 $\rm deg^2$ dark matter lightcone from the Uchuu cosmological simulation with the simulated infrared dusty extragalactic sky (SIDES) approach. We find that in order to constrain the CO LF with an uncertainty below 20%, we need survey sizes of at least 0.1 $\rm deg^2$. Furthermore, accounting for the field-to-field variance using only the Poisson variance can underestimate the total variance by up to 80%. The lower the luminosity is and the larger the survey size is, the higher the level of underestimate. At $z$<3, the impact of field-to-field variance on the cosmic molecular gas density can be as high as 40% for the 4.6 arcmin$^2$ field, but drops below 10% for areas larger than 0.2 deg$^2$. However, at $z>3$ the variance decreases more slowly with survey size and for example drops below 10% for 1 deg$^2$ fields. Finally, we find that the CO and [CII] LIM power spectra can vary by up to 50% in $\rm 1 deg^2$ fields. This limits the accuracy of the constraints provided by the first 1 deg$^2$ surveys. The level of the shot noise power is always dominated by the sources that are just below the detection thresholds. We provide an analytical formula to estimate the field-to-field variance of current or future LIM experiments. The code and the full SIDES-Uchuu products (catalogs, cubes, and maps) are publicly available.