Was there any opposition to placing the images of American presidents on currency -- perhaps out of fears that it too-closely echoed the practices of royalty?

Yes, there was. Most early American coins and Federal Notes usually had an allegorical/mythological figure figure or symbolic animal on them, rather than a former leader.

For example, from 1793 to 1857, the "large cent" coin (predecessor to the modern penny) had an female head or bust, intended as an allegorical image of "Liberty". The wiki page has good pictures.

A similar neoclassical "Liberty" female figure was the single common design element on American money until well into the 19th century. Idealized "Indians" were also common, as were animals, especially and Eagle similar to the one depicted on the Great Seal of the United States. There was also the Buffalo Nickel minted in the later 19th century, which as the name may suggest, featured an American Bison on the obverse.

Because it was the universal practice, dating back to antiquity, for coins to feature the visage (usually in profile) of the current monarch, the U.S. Mint made a deliberate effort NOT to put an actual person or ruler, living or dead, on the coinage. The first major exception to this was the introduction of the Lincoln Cent in 1909. This was originally intended as a limited edition commemorative (1909 was the centennial of Abraham Lincoln's birth), but proved so popular that it has remained in production (with variations) ever since.

The practice of not depicting living persons was more of a norm than an explicit rule, mostly. In the 20th century there were several attempts to depict a living person or their relatives on a commemorative (notably sitting president Coolidge on the 1926 sesquicentennial half dollar). There efforts often provoked political controversy, and generally the Mint has avoided doing it except sometimes on limited-circ commemoratives.

However, it currently (since 1995?) mandated by federal law that no living person can appear on the coinage, and even former presidents must have been deceased for at least two years before they can be depicted in the modern "Presidential Dollar" series.

This web page by coin collector site NumisMaster explains the recent history and controversies regarding living persons on U.S. coinage. A version of this article appeared in the Numistatic News (a periodical) sometime in the last few years, I think. I will attempt to find a more permanent citation if anyone wants one.