The general european catholic resentment for this alliance can be easily picked up by reading the wikipedia article. However, what were the consequences of this alliance for the Ottoman diplomacy in Asia?
Did muslim rulers have some kind of ideological, principled opposition to an alliance "between the Lily and the Crescent"?
Was there any notable skepticism, opposition or outright hostility to this alliance among the Ottoman court? If so, did it tone down over the 200 years the alliance lasted?
I don't know about any specific Muslim criticism from a religious viewpoint regarding the alliance, so I can't comment on this.
The political consequences of the alliance in Asia were not as grave as in Europe. This is firstly due to the general anti-Habsburg nature of the Franco-Ottoman agreement with had primarily consequences in the Mediterranean and Central European theatre. And secondly due to the fact that France in the 16th and 17th century was not able to sufficiently project power into South Asia so that she could be of considerable help for the Ottomans there.
However it did have some interesting implications: The largest Asian adversary of the Ottomans, Safavid Persia, did reach out to European powers several times to coordinate moves against her larger Sunni enemy. Already in the early 16th century contact was established between the Safavids and the Portuguese as the latter were new to the Indian Ocean and eager to eliminate the Ottoman danger to their fledgling trade network, but it did not prove to be of larger consequence. Then, during the reign of Charles V./I., monarch of Spain and the Holy Roman Empire in personal union, the Habsburgs tried to coordinate attacks on the Ottoman Empire with the Persians. Their agreement (in 1524 if I remember correctly) was actually made before the signing of the Franco-Ottoman alliance which was formalized not until 1536. However due to extreme distances and very different geopolitical situations (the Persians had to deal with many other opponents in their own eastern domains) coordination proved to be difficult. Safavid Persia went on to send two more important delegations to Europe: The embassy of 1599-1602 and the embassy of 1609-1615 (bear in mind that this was a long and troublesome voyage in the early 17th century) which were well received in Habsburg courts.
The Habsburg-Persian alliance (if we were to even speak of one) never matured to a point comparable to that of the Franco-Ottoman alliance. Geography is a huge factor in this, I think. Three of the four players in this (geopolitical) game were situated around the Mediterranean (from west to east: France, the Habsburg domains, the Ottoman Empire) while the last one (Persia) was far on the other side of the events, separated by the entire Middle East under control of the Ottomans. Sharing the close Mediterranean theatre, tightly knit by maritime trade routes since ancient times, communication and coordination of French and Ottoman forces was comparatively easy, while the Habsburgs and the Persians shared nothing of the sort.