Protesters waved the national flag and chanted "Tebboune is not our president" and "we will continue our protests" in a heavily policed central district of the capital Algiers.
Some also chanted: "Police are traitors" while others called for people in the western city of Oran, where there have been reports of arrests, to keep on marching.
The protesters have marched every Friday since February, demanding that the entire ruling elite quit power, an end to corruption, and that the army withdraw from politics.
In April, the unrest forced veteran president Abdelaziz Bouteflika to step down, as the military and its allies purged his cronies in an anti-corruption drive while pushing for new elections.
Protesters rejected any election as illegitimate so long as the old ruling elite, including the military, held power. Tebboune was elected with 58% of the vote last week, but with even official figures showing only 40% turnout.
He and the four other candidates, all former senior officials, praised the protesters for seeking a patriotic renewal of Algerian politics, but without acceding to their demands for a more thorough purge of the hierarchy.
At his inauguration on Thursday, Tebboune offered dialog with the protesters and pledged to start work soon on a new constitution that would limit presidents to two terms in office.
However, he also presented a medal to army chief Lieutenant General Ahmed Gaed Salah, whose resignation protesters have demanded for months.The mass protest movement has no formal organization or leadership, but some prominent supporters have urged dialog with Tebboune. Demonstrators on Friday rejected talks with a man they regard as a puppet of the old ruling elite.