One that I can think of is the memorial marker for the Mountain Meadows Massacre in SW Utah.

The massacre itself happened from September 7th through the 11th 1857, when a group of local Mormons slaughtered over 120 Arkansas immigrants on their way to California. This led to years of federal investigations and several trials before a single man, John Lee, was found guilty of the massacre in 1876 and executed. In a true conspiracy of silence, the vast majority of the perpetrators got away with it.

In May 1859 a US Calvary unit led by Major James Carleton arrived at the site. They buried the remains of at least 30 adults that were still laying about the site. Then they erected a 24 tall monument to the massacre with the words "Vengeance is mine, I will repay saith the Lord," carved into the side.

Two years later in May 1861 Brigham Young and a group of Mormon leaders toured the site. Young is quoted as saying "Vengeance is mine, and I have taken a little." He then ordered the monument to be dismantled and the pieces of it scattered around the site.

A few years later in May 1864 (before you ask, no, I don't know why everyone loved to visit the site in May) Captain George Price visited the site with another company of cavalry and found that the monument had been torn down. He and his men spent some time rebuilding the monument, though smaller than before. This replacement monument was only 14 feet tall.

It was routinely defaced. Sometime around 1874 it was dismantled again, only to be rebuilt shortly again after.

By 1932 it had again been largely torn down and defaced. In response the area around the monument was enclosed by a 4' tall rock wall with a brass memorial plaque attached to the wall. This was completed and dedicated in August 1932. The 1859/1864/1874 monument was left in its partially destroyed state.

In 1965 the land containing the memorial was donated to the LDS church. They didn't touch either the original monument itself, or the surrounding wall. But they removed all signs to the site, and let the road deteriorate so it became impossible to drive to the memorial. The local county ended up taking control (and maintenance) of the road, and reinstalled the signs leading to the site.

In 1990 a new monument was constructed by the descendants of the survivors of the massacre on Forest Service land so as to have a monument to the event on land not controlled by the LDS church. At the same time this new monument was constructed the old 1932 monument was also cleaned and repaired by the church.

There have been further developments around the site and the monuments, but they cross the 20 year rule and as such aren't appropriate to this forum. The final chapter around the MMM site and now multiple monuments and memorials,has yet to be written.

The fights over the various monuments have had both a lot to do with the fact that the original perpetrators and their descendants have always lived in proximity to the site. They generally haven't been very happy to acknowledge the unprovoked murder of men, women, and children. Likewise the LDS church has been closely linked to the massacre. John Lee claimed that Young and other church leaders ordered the attack. There is no doubt that Young and others were complicit after the fact, shielding Lee and other perpetrators for decades. They spent decades trying to remove the massacre and the various monuments about it from memory.

It's not a monument, but you might be interested to hear about the controversy surrounding Tilted Arc by the American post-Minimalist sculptor Richard Serra. The large sculpture, a picture of which you can see here, was commissioned by the United States General Services Administration as part of its Arts-in-Architecture program for the plaza in front of a federal building in Manhattan. Serra was already known for his large-scale sculptures, usually constructed out of large sheets of rolled steel, so he perhaps seemed like a good fit for a site-specific work. However, the sculpture was almost immediately controversial, owing to the fact that it basically blocked off the entire plaza. Employees chafed at having to walk around it when they wanted to cross the plaza, and many people raised safety concerns, both because muggers and other predators might potentially hide behind it and because it might potentially be used by terrorists to deflect explosions (or perhaps even used as part of an explosive device--though it's worth nothing that this sucker was HUGE, so you'd need a lot of explosive to make it budge). This was before both of the World Trade Center and Oklahoma City bombings, but the 1970s were an era where a lot of terrorists were setting off bombs, so this is not a totally outlandish concern, though the thing was really pretty huge.

Anyway, many of the employees of the building signed a petition to have it removed, and a number of judges and other officials made public statements opposing it. On the other side were arrayed a number of art world types, including critics and artists, as well as Serra himself. Serra stated unequivocally that the work was site-specific and to remove it was to destroy it. Eventually, however, there was a lawsuit to remove the piece, and even though such art world luminaries as Philip Glass and Keith Haring testified in favor of its remaining, it was removed in 1989. They shipped it to a parking lot in Brooklyn, but has more recently been moved to a warehouse facility in Maryland. The federal government still owns the steel, but since Serra is adamant that the piece cannot function outside of its intended location, it will probably remain there forever.

The aftermath of the controversy saw a couple of results. First, the General Services Administration scaled back its art purchasing program, tending towards safer and less potentially controversial work in the future. This tendency was only reinforced after the Mapplethorpe/Serrano/NEA funding controversies of the 1990s. Additionally, in 1990 Congress passed the Visual Artist Rights Act, an amendment to the Copyright Act of 1976, designed to specifically safeguard the rights of the artist if a similar situation ever arose again. Specifically, it safeguards what are called moral rights, and is the first statute in the United States to do so. It gives artists certain rights over their work even after it has been sold, including defense against misattribution, as well as the right to prevent misuse, damage, or destruction of a work of art if such an action would damage the artist's reputation. One of Serra's big complaints about the whole affair was that it privileged property rights over his right to freedom of expression, and so this law was designed to guarantee certain protections to future artists who accepted public commissions.

So it isn't a monument, but it is a major, site-specific piece of artwork that was removed after a lawsuit. Note that the moral rights afforded by VARA only last the lifetime of the creator, so I doubt it would cover many of the Confederate monuments out there.