In searching for images of photos taken during construction of the monument, nothing's turned up that corroborates this. Although in general during that era, photos probably weren't taken often, and may have been framed in such a way to not show a tree in the middle to keep the monument itself the subject. And of course, some images may have been scrubbed or search excluded.
First I've heard this, interesting though.
https://www.youtube.com/results?search_query=washington+monument+tree+wormhole
In searching for images of photos taken during construction of the monument, nothing's turned up that corroborates this. Although in general during that era, photos probably weren't taken often, and may have been framed in such a way to not show a tree in the middle to keep the monument itself the subject. And of course, some images may have been scrubbed or search excluded.