A comprehensive study on the mechanical properties of a pure carbon nanotube (PCNT)/nitrile butadiene rubber (NBR) composite and an amide-functionalized carbon nanotube (CONH2–CNT)/nitrile butadiene rubber (NBR) composite was carried out using molecular dynamics (MDs) simulations at different temperatures. The effects of temperature on the mechanical properties, fractional free volume (FFV), MSD, dipole autocorrelation function, number of hydrogen bonds of PCNT composites, and functionalized CNT composites were analyzed and compared, and the pull-out behavior of the composites under different condition temperatures was simulated. The enhancement mechanism of the interface interaction between the functionalized carbon nanotubes and the NBR matrix was explained from an atomic point of view. The results show that, due to the existence of hydrogen bonds, higher interfacial binding energies were formed between PCNT and NBR, and FFVs and MSDs were restricted at each temperature, with the mechanical properties of the composites being improved by 5.02–25.93%.