The intensification of global climate change has made the study of greenhouse gas (GHG) emissions increasingly important. To gain a deeper understanding of the emission characteristics and driving factors of nitrous oxide (N2O), carbon dioxide (CO2), and methane (CH4) from rubber plantation soils, this study conducted a 16-month continuous observation in a rubber plantation in Danzhou, Hainan, employing the static chamber method for the monthly sampling and measurement of GHG emissions while analyzing the soil’s physical and chemical properties. The results indicated that the N2O flux exhibited no significant diurnal variation between the dry and rainy seasons, with an average emission rate of 0.03 ± 0.002 mg·m−2·h−1. A clear seasonal trend was observed, with higher emissions in summer than in winter, resulting in an annual flux of 3 kg·hm−2·a−1 (equivalent to 1.9 kg N·hm−2·a−1). N2O emissions were significantly correlated with soil temperature and moisture, explaining 46% and 40% of the variations, respectively, while soil ammonium nitrogen content also significantly influenced N2O and CO2 emissions. The rubber plantation soil acted as a source of N2O and CO2 emissions and a sink for CH2, with lower emissions of N2O and CO2 during the daytime compared to nighttime, and higher CH4 uptake during the daytime. In the dry season, there was a significant positive correlation between N2O and CO2 emissions (R2 = 0.74, p < 0.001). This study reveals the diurnal and seasonal patterns of GHG emissions from rubber plantation soils in Hainan and their interrelationships, providing a scientific basis for the low-carbon management of rubber plantations and GHG mitigation strategies, thereby contributing to attempts to reduce the impact of rubber cultivation on climate change.