Problem solved!

I'm not sure what solved it, so here's a list of what I did:

-uninstall Qt, remove any remaining files by hand
-reinstall Qt
-edit the g++ mkspecs (add -static to lflags)
-configure -static -release -no-qt3support -no-exceptions -no-qmake -fast
-make sub-corelib,sub-gui,sub-network,sub-svg, and sub-plugins (maybe I forgot one)

My .pro file's relevant lines are now

QT += svg
QTPLUGIN += qsvg
LIBS += -L$$[QT_INSTALL_PLUGINS]/iconengines
LIBS += $$[QT_INSTALL_PLUGINS]/iconengines/libqsvg.a

With this, my program compiles and shows the icons when run.

Thanks!