Alexander Ankvab reportedly retreated to his home town Gudauta, 40km (25 miles) away, when talks with the opposition broke down.
Many people in the Russian-backed region are unhappy with the ailing economy and lack of reforms.
But Abkhazian nationalist feeling is also fuelling the unrest.
By Michael CecireForeign Policy Research Institute
The 2014 Sochi Olympics were billed as a way of boosting the economy by bringing tourism to Abkhazia or sourcing materials for construction, but the most noticeable effect was actually a source of frustration: the expanded security cordon that penetrated beyond the Russia-Abkhazia border.
Some economic problems can certainly be attributed to the de facto government, but it is Russia that has the real power to improve conditions. For example, the Moscow-financed building boom in Chechnya has improved material conditions (even if human rights remain a deep problem), but nothing comparable has happened in Abkhazia.
However, the opposition's chief point of contention is Alexander Ankvab's more liberal (relatively speaking) policy towards ethnic Georgians in Abkhazia. He has advocated more fully integrating them, granting them Abkhazian "citizenship". The opposition has painted this as a purely political ploy to expand his electoral base and say it could endanger Abkhazia's ethnic Abkhazian identity.
Analysis: Unrest in Abkhazia
The protesters are led by Raul Khadzhimba, a former prime minister and vice-president of Abkhazia, who was defeated by Mr Ankvab in elections in August 2011.
The Black Sea coastal region is recognised as a state only by Russia and a few other countries. It broke away from Georgia after a civil war more than 20 years ago and declared formal independence in 2008. Since then, it has relied militarily and financially on Russia.
Russia is reportedly sending two senior officials, presidential aide Vladislav Surkov and deputy security council secretary Rashid Nurgaliyev, to Sukhumi in response to the crisis.
Opinion among the opposition is said to be divided between those who want Abkhazia to rely less on Russia and those who instead want it to become part of Russia.
Mr Ankvab accused the opposition of attempting a coup on Tuesday and said the security forces remained "loyal to the state" and were "taking measures to stabilise the situation".
Raul Khadzhimba defended the protests, saying: "Over all the years of his rule the president did not allow anybody in his circle to do their work.
"He took upon himself everybody's responsibilities on their behalf, whether he should or should not have been doing this. That led to our country in fact becoming an authoritarian regime.
In a statement, the Russian foreign ministry said: "The Russian side is following events closely and with concern... and considers it important that socio-political processes develop exclusively along legal lines."
The president of the breakaway Georgian region of Abkhazia is said to have fled the capital Sukhumi after opposition protesters seized his office.