High-resolution observations of (CO)-C-12 (2 ''.3), (CO)-C-13 (3 ''.9), and HCN (5 ''.4)J = 1-0 in the galaxy merger Arp 299 (IC 694 and NGC 3690) show that the line ratios vary dramatically across the system. The (CO)-C-12/(CO)-C-13 ratio is unusually large, 60 +/- 15, at the IC 694 nucleus, where (CO)-C-12 emission is very strong, and much smaller, 10 +/- 3, in the southern extended disk of that galaxy. Elsewhere, the (CO)-C-12/(CO)-C-13 line ratio is 5-20, typical of spiral galaxies. The line ratio variation in the overlap between the two galaxies is smaller, ranging from 10 +/- 3 in the east to 20 +/- 4 in the west. The (CO)-C-12/HCN line ratio also varies across Arp 299, although to a lesser degree. HCN emission is bright toward each galaxy nucleus and in the extranuclear region of active star formation; it was not detected in the IC 694 disk or the eastern part of the overlap region, leading to lower limits of 25 and 20, respectively. By contrast, at the nuclei of IC 694 and NGC 3690 the ratios are 9 +/- 1 and 14 +/- 3, respectively. In the western part of the overlap region it is 11 +/- 3. The large (CO)-C-12/(CO)-C-13 1-0 intensity ratio at the nucleus of IC 694 can primarily be attributed to a low-to-moderate optical depth (T less than or similar to 1) in the (CO)-C-12 1-0 line. These data support the hypothesis that unusually high (CO)-C-12/(CO)-C-13 line ratios (> 20) are associated with extremely compact molecular distributions in the nuclei of merging galaxies. Relative to (CO)-C-12, (CO)-C-13 1-0 is brightest in quiescent regions of low (CO)-C-12 surface brightness and weakest in starburst regions and the galactic nuclei. A medium consisting of dense (n = 10(4)-10(5) cm(-3)) and warm (T-k > 50 K) gas will reproduce the extreme line ratios observed in the nucleus of IC 694, where the area filling factor must be at least 20%.