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We study the effect of baryonic processes on weak lensing (WL) observables with a suite of mock WL maps, the $κ$TNG, based on the cosmological hydrodynamic simulations IllustrisTNG. We quantify the baryonic effects on the WL angular power spectrum, one-point probability distribution function (PDF), and number counts of peaks and minima. We also show the redshift evolution of the effects, which is a key to distinguish the effect of baryons from fundamental physics such as dark energy, dark matter, and massive neutrinos. We find that baryonic processes reduce the small-scale power, suppress the tails of the PDF, peak and minimum counts, and change the total number of peaks and minima. We compare our results to existing semi-analytic models and hydrodynamic simulations, and discuss the source of discrepancies. The $κ$TNG suite includes 10,000 realisations of $5 \times 5 \, \mathrm{deg}^2$ maps for 40 source redshifts up to $z_s = 2.6$, well covering the range of interest for existing and upcoming weak lensing surveys. We also produce the $κ$TNG-Dark suite of maps, generated based on the corresponding dark matter only IllustrisTNG simulations. Our mock maps are suitable for developing analytic models that incorporate the effect of baryons, but also particularly useful for studies that rely on mass maps, such as non-Gaussian statistics and machine learning with convolutional neural networks. The suite of mock maps is publicly available at Columbia Lensing (http://columbialensing.org).