Modified from an earlier answer of mine

The border regions of Alsace-Lorraine were where the bulk of France's Jews lived prior to 1789. Antisemitic laws and restrictions as well as expulsions had whittled the community outside of these border regions. Even within this region, antisemitic restrictions created a degree of social segregation in which Jews lived in a social world apart from their gentile neighbors. The result is that a good deal of Alsace-Lorraine's gentile leaders not only had experience with the practical application of antisemitic laws, but tended to see Jews as aliens or a foreign element.

That was different mentality than the few members of the National Assembly who deigned to grant Jews rights. These proposed measures were not exactly emancipation or acceptance of Judaism within France, but rather treated Jews as individuals who would gradually shed their Judaism under the new regime. The Declaration of the Rights of Man and Citizen did in theory call for emancipation of Jews and the elimination of antisemitic restrictions for all citizens of France, extending these rights to Jews was not something that the revolutionaries considered when drafting this document. Yet the logic of the Declaration and other social leveling projects of the Revolution prompted Paris to turn to the matter of the approximately 40000 Jews within France. The Revolution fostered the elimination of corporate privileges of the Jewish communities and the tentative first steps towards accepting Jews as full citizens of France. The moderate liberal Stanislas Clermont-Tonnerre would argue in the National Assembly that:

We should refuse the Jews any rights as a separate nation, but grant them the same as individuals. They must not make in the State either a distinctive corps or an order. They must become individual citizens.

Similarly, the revolutionary firebrand Henri Grégoire published a pamphlet in October 1789 favoring Jewish emancipation. Grégoire's motions for Jewish emancipation received tentative support from other political actors like Mirabeau and Robespierre. In the latter case, the Jacobin leader saw the lack of Jewish integration into French culture as baleful legacy of the monarchical regime:

The vices of the Jews have their roots in the degradation into which you have plunged them. They will be good when they see some benefit in being so.

Like many other gentile contemporaries sympathetic toward Jewish citizenship, Robespierre took various antisemitic tropes such as usury and corruption as a given. Yet the chaos of the Revolution's later stages meant that emancipation, which passed in the National Assembly in 27 September 1791, would be highly patchwork and incomplete.

The Alsatian deputies formed the core group resisting against these measures and this dovetailed with their largely conservative politics that was not atypical for deputies from border regions. The Bishop of Nancy and one of the leaders of the moderate right in the National Assembly, Anne-Louis-Henri of La Fare, argued against Clermont-Tonnerre's motion for Jewish citizenship by arguing that the Jews were not Frenchmen, but foreigners that deliberately segregated themselves from their local gentile community. The Alsatian Jean-François Reubell, another voice on the right, was vocal in his opposition to emancipation, claiming "Who can assure us that we won't be required to have ourselves all circumcised before we're thirty?" Other political opposition figures both within the National Assembly and without would use the Revolutionary government's support for emancipation as a political wedge issue to argue that the Revolution was serving the interests of outsiders like protestants and Jews.

The other pushback against emancipation came from within the Jewish community itself. While a good many Jews welcomed full legal rights, they looked askance at the abolition of local corporate privileges which they perceived was yet another attack on Jewish institutions. While many Jewish communities engaged in symbolic gestures of fidelity to the Republic, they often couched them in terms that made it clear the Jewish community was still a united one. Although they framed their support for the Revolution in diplomatic terms, Jewish leaders supported the federalist faction of the Girondins, which called for a degree of local autonomy. Jewish community leaders' caution about the Revolution was justified by the resurgence of popular antisemitism that came about with the outbreak of war. In the frontier regions, rumors spread that the Yiddish-speaking Jews were aiding the enemies of France. Although the economic chaos inflation and the sale of church lands actually hurt Jewish economic interests, this did not stop the popular and ancien regime practice of blaming Jews for economic woes. The new state authorities often did not take complaints of popular violence and pogroms seriously, which further underscored the skepticism about the intentions of the new regime. The cautious support for the moderate federalist wing of the Revolution also drew the ire of some of the radical Jacobins; for example, the Jacobin radical Marc Antoine Baudot would cite Jewish support for Federalists as necessitating "a regeneration for the Jews guillotine style."

Napoleon was in many respects the culmination of the Revolution's Jewish policy. Although the National Assembly had granted Jews full citizenship rights towards the end of its existence, the new revolutionary state also sought to subordinate and systematize Jewish religious and social institutions. Napoleon, who was a systematizer par excellence, convened a council of Jewish elders, the Grand Sanhedrin, to declare that Jewish conscription was religiously valid. This followed other governmental attempts to regulate its Jewish community, whether through dress or the state control over rabbis. Napoleon ushered in the French government's century-long attempt to subordinate religion to the state oversight and control. Berr Isaac Berr de Turique, the Alsatian Jewish delegate to the National Assembly, would sum up these initiatives in an 1806 pamphlet:

Hardly had the decree which granted Jews the title of citizen been handed down than they were reproached with not yet having become worthy as a group for this great favor.

Berr's pamphlet prophesied many of the problems that Jews in France would face after the Revolutionary and Napoleonic Era. Even a supporter of Jewish rights like Clermont-Tonnerre had a relatively narrow definition of what exactly constituted a Jew and saw that Jewishness as a collective identity as antithetical to the new French citizen the Revolution was going to create. The systematizing and homogenizing processes did turn a number of Jews into Frenchmen over the long term, but the larger acceptance of social antisemitism acted as a counterweight to a total integration. Sometimes this discrepancy between legal and social realities would explode, as in the case of the Dreyfus Affair.

Sources

Aston, Nigel. Religion and Revolution in France, 1780-1804. Washington, D. C.: CUA Press, 2000.

Hyman, Paula. The Jews of Modern France. Berkeley: University of California Press, 1998.

Tozzi, Christopher. "Jews, Soldiering, and Citizenship in Revolutionary and Napoleonic France." The Journal of Modern History 86, no. 2 (2014): 233-257.