The Bible tells us that blood should be offered with the flesh in burnt offerings:
"And you shall offer your burnt offerings, the flesh and the blood, on the altar of the Lord your God; and the blood of your sacrifices shall be poured out on the altar of the Lord your God, and you shall eat the flesh." (Deuteronomy 12:27)
So some amount of blood was burned up.
According to Jewish tradition in the Mishnah (a collection of Jewish rabbinic oral traditions, completed around 200 AD), the altar and its ramp were cleaned every Friday with a cloth to remove accumulated blood.
According to the Mishnah, the second Temple had the following features:
"At the southwestern corner [of the foundation] there were two openings like two small nostrils through which the blood which was poured on the western side of the foundation and on the southern side flowed down till the two streams became mingled in the channel, through which they made their way out to the Kidron wadi. The blood of all sacrifices was either drained on the western side of the foundation or on the southern side. The blood would drain out through two holes shaped like nostrils, and then flow down to the channel that flowed through the Temple Courtyard. From there the blood would be flushed out to the Kidron wadi that flows below the Temple Mount." (Middot 3:2).
So the altar had two openings at its base that drained blood into underground channels. This blood flowed through a system of underground conduits into the Kidron Valley. This system prevented blood from pooling and ensured a continuous flow away from the Temple area, especially during high holy days when many sacrifices were made.