How does the mood of "Introduction to Poetry" change beginning in line 12? What do you make of the shift from "them" to "they"?

Starting with line 12 and lasting to the end of the poem, the mood becomes much darker, more somber, and more unpleasant.
In the first twelves lines, the poem is light and playful as the speaker apparently describes how he encourages his students to interact with a poem. (It is important to note that he never states he is the teacher and they are the students; however, whether or not they are his students in a formal classroom setting, they are his "students" in the sense that he is trying to teach them something about poetry.)
In the first part of the poem, the speaker urges the students to be experimental, sensory, creative, and interactive with a poem, to approach it in new ways so as to "feel" it. He wants them to treat a poem as a physical space with dimensions: he says they should drop a mouse into it and watch what happens, or feel its walls for a light switch, or waterski across it. All of these tactile images point to being interactive, to feeling the words of a poem in our bodies.
With the shift from "them" in stanza five to "they" in stanza six, the narrator suddenly distances himself from his students. In the first five stanzas he uses the pronoun "I," showing he is trying to interact with "them." By stanza six, he has withdrawn and is watching them—the "they"—from afar as, against his advice, they torture a poem. The most important shift, to my mind, is the way the narrator drops the "I" and simply observes at the end what the students are doing:

But all they want to do is tie the poem to a chair with rope and torture a confession out of it. They begin beating it with a hose to find out what it really means.

In these last lines, the speaker watches with dismay as his students (or possibly society as a whole) ignore his enthusiastic words about interacting with a poem creatively and experimentally. Now the images are distasteful—tying a poem to a chair with a rope, torturing it, beating it with a hose. We want to turn away from the pictures these words conjure in our minds. Collins is telling us that the detached, intellectual way we normally try to finding meaning or theme in a poem violates and harms a poem. When try to control a poem, rather than simply be with it, we destroy or maim it.