Map from Google - Burundi

In Burundi, a country located just south of the equator, the climate is mild or warm, and varies according to altitude, while there is little seasonal variation in temperature. The majority of the country is occupied by a plateau, at an altitude of around 4,900/5,900 feet, but in the west there is also a lower region, at around 2,600 feet, as well as a mountainous belt.
From June to August, the austral winter, dominated by cool breezes from the south-east, is a dry season, in which there is little rain and the sun often shines. From mid-August, the temperatures begin to rise, and in September and October the highest temperatures of the year occur; at the same time, however, the rains begin, falling in the form of downpour or thunderstorm, and they gradually lower the temperature, which returns to the values of the rest of the year, around 72/77 °F during the day on the plateau, and around 81/82 °F in the western lowlands.
If we exclude the three dry months, moderate rains fall in the rest of the year. The rains tend to increase with altitude, ranging typically from 47/55 inches per year in the plateau, to below 40 inches in the western area, which is the warmest and least rainy of the country. In this area we find Lake Tanganyika, whose banks are located at 2,500 feet above sea level. The lake mitigates the climate especially at night, keeping temperatures somewhat higher, and causing a bit of sultriness, but bringing also beneficial breezes during the day.
The capital Bujumbura is located along the shores of the lake. Here it's hot throughout the year, but without excesses, given that the maximum temperatures are around 84 °F all year round, except from mid-August to October when they reach 86/88 °F.
Here are the average temperatures.
Average temperatures - Bujumbura
BujumburaJanFebMarAprMayJunJulAugSepOctNovDec
BujumburaJFMAMJJASOND
Min (°F)666666686664636366666666
Max (°F)848684848686848688868484

On average, 33 inches of rain per year fall, peaking in April and May, when the rainfall amount reaches or exceeds 3.5 inches per month, and a minimum from June to August, when the rains are rare and sporadic. Here is the average precipitation.
Average precipitation - Bujumbura
BujumburaJanFebMarAprMayJunJulAugSepOctNovDecYear
BujumburaJFMAMJJASONDY
Prec. (in)3.54.34.74.92.20.40.30.41.22.43.94.532.8
Days11121310611146131290

The altitude drops below 3,300 feet also in the west, along the Rusizi River, where it can be hot between September and March, with peaks around 95 °F.
In the western part, but east of the Rusizi River, there is also a mountainous area, where the altitude exceeds 6,500 feet. The northern part is occupied by the Kibira National Park, while in the south we find Mount Heha, 8,806 feet high, which is the highest peak of the country.
Here are the average temperatures of Rwegura, located at 7,500 feet above sea level.
Average temperatures - Rwegura
RweguraJanFebMarAprMayJunJulAugSepOctNovDec
RweguraJFMAMJJASOND
Min (°F)545454545252525254545454
Max (°F)707068686668687070706870

In this mountainous area, rainfall exceeds 64 inches per year, and is abundant from October to April, although there is a certain decrease in January and February. Here is the rainfall in Rwegura.
Average precipitation - Rwegura
RweguraJanFebMarAprMayJunJulAugSepOctNovDecYear
RweguraJFMAMJJASONDY
Prec. (in)6.15.98.19.43.910.21.4110.27.79.364.4
Days201822241742413202322189

The sunshine amount in Burundi is good in the dry season, especially in June and July, while it becomes just acceptable from November to March. A bit of cloudiness can occur in all seasons, especially at higher altitudes.

Bujumbura

When to go

The best time to visit Burundi runs from June to August, being clearly the most dry and sunny. Anyway, in this season nights can be cool or even cold, with minimum around 54 °F in Bujumbura, and even below on the plateau.

What to pack

In winter (June to August). In Bujumbura, Lake Tanganyika and southern plains: light clothes, a sweatshirt and a light jacket, a light scarf for the breeze. In the plateau and the mountains, spring/autumn clothes, light for the day, sweater and warm jacket for the evening, sun hat, hiking shoes.

In the rest of the year. In Bujumbura, Lake Tanganyika and southern plains, light clothing, a light sweater for the evening, light raincoat or umbrella. In the plateau and the mountains, spring/autumn clothes, being ready to remove the outer layer during the day, jacket, raincoat for the evening; sun hat, hiking shoes.