Do you care about potential spoilers?
If you don't care, continue reading as by the end of the Forerunner-Flood War. Only a handful of Forerunners were alive on the ark after the Halo's firing and while the number wasn't specified. Multiple rates were known to have survived: Warrior-Servants, Life Workers, Builders, Miners and the plasma jockey enginneers (not the flying squid in Halo 3 ODST). After the war it is stated the Forerunners discarded their "rates" and solely focused on the Conservation Measure, returning life to galaxy. Once that was completed, the Iso-Didact now going by his old name Bornstellar led a journey to Maethrillian where they "reset" the domain at the cost of more Forerunner lives.
Many side characters introduced throughout the Forerunner Saga are judged and killed by a Precursor A.I.
After the mission was completed we know the surviving Forerunners which include Bornster-Makes-Eternal-Lasting and Chant-to-Green with a unspecified amount of Forerunners departed the galaxy in exile. Eventually they would settle in a new galaxy where the Forerunners scattered themselves thin so that in time they would go extinct. Bornstellar and Chant settled on a planet and chose a rather primitive lifestyle without their advanced technology and eventually they had a son together. It isn't known what happened to the rest, but I personally believe some may have banded together in hopes of rebuilding Forerunner society and if that is the case, I hope to see their descendants one day.
Now besides the exiled Forerunners and the Ur-Didact, a unknown Builder also survived the firing of the Halo Array.
Halo 5 Guardians introduces us to this unknown builders presence in the terminals and he was arround when Cortana's uprising began. It isn't known how he survived, but it is apparant he wasn't part of the Conservation Measure. The unnamed builders goal is to find a location that is only known by name as "Bastion" and that may or may not be a shield world where other Forerunners may have survived. Without access to the domain for almost 100,000 years the builder searched in vain across the galaxy as "Bastion" was noted to have been moved which would have incurred a great amount of slipspace reconciliation and before the end of Halo 5, the builder was finally able to find Bastion's location.
Now onto your question of if any Forerunners are still alive.
Discounting the Ur-Didact who is composed and the unnamed builder who managed to escape Cortana's clutches. The Forerunners in the Milky Way Galaxy are effectively extinct. Their is a possibility they continued as a species in the new galaxy mentioned above and their is the sub-species mentioned in the Magellanic Cloud in Halo Silentium who could have survived possibily as well. They were introduced in Halo Silentium and were noted to bear very similar traits to humans in contrast to their brethern in the Milky Way. It is implied though they were killed after Omega Halo fired at the end of the war. I have hopes they survived though as one halo shouldn't have been enough to wipe everything.
Last piece of information.
The monitor of Installation 00; Tragic Solitude was the Forerunner Splendid-Dust-of-Ancient-Suns. He was the First Counser of the Ecumene council and he alone shouldered the knowledge of the Forerunners dark secrets hidden within Maethrillian by the end of the war. He knew the mythical organon that Bornstellar was searching for was in fact a Precursor A.I. buried deep in the vaults of Maethrillian. After journeying to the capital and returning with scars that would never fade, he chose to become the custodian/monitor of Installation 00 so that when humanity finally arrived, he would be able to guide them in hopes that they wouldn't repeat history.
Unfortunately time wasn't his ally.
After becoming a monitor his memories as Splendid-Dust were erased and the 100,000 years of solitude eventually led to his rampancy. The final battle of the Human-Covenant War on the Ark in 2552 didn't help things either as he wanted to purge the galaxy by the end of it and nearly succeeded, but in a tragic way. One could say Tragic Solitude probably found peace in death as in the case of 343 Guilty Spark. The memories of his past as Chakas returned after his carapace was damaged and right before his death. I personally believe the memories of Splendid-Dust returned only in that moment and with them, his eternal vigil came to an end.
I personally hope Forerunners are still alive in either "Bastion" or the new galaxy though.