So how do modern "emotional" worship songs botch up theology? Maybe that's too broad of a question. Name 5 modern worship songs that God would hate. Personally, I like two hymns very much, How Great Thou Art, and It Is Well with My Soul, and they were totally emotional songs when penned.
Try "No Longer Slaves" from Bethel (my thoughts in parentheses after each section)
You unravel me with a melody
You surround me with a song
Of deliverance from my enemies
'Til all my fears are gone
(What does this even mean? You unravel me with a melody? Huh?)
I'm no longer a slave to fear
I am a child of God
I'm no longer a slave to fear
I am a child of God
(This part isn't a problem)
From my mother's womb
You have chosen me
Love has called my name
I've been born again to a family
Your blood flows through my veins
(Your blood flows through my veins? This is either a metaphor for the sake of rhyming or it's simply not true)
I'm no longer a slave to fear
I am a child of God
I'm no longer a slave to fear
I am a child of God
I'm no longer a slave to fear
I am a child of God
I'm no longer a slave to fear
I am a child of God
You split the sea
So I could walk right through it
My fears are drowned in perfect love
You rescued me
And I will stand and sing
I am a child of God
(God certainly has rescued us but I don't remember the sea being split open very often. I don't know about anyone else but I still have fears - it's a nice soothing thought that they can be drowned in perfect love but unless someone wants to claim they never feel any fears at all I'm not sure this line works either)
As far as emotional responses go, we can even take a verse that's theologically perfectly sound and turn it into more of an emotional experience than anything else. Take the verse from "In Christ Alone":
There in the ground his body lay
Light of the world by darkness slain
Then bursting forth in glorious day
Up from the grave he rose again
And as he stands in victory
Sin's curse has lost its grip on me
For I am his and he is mine
Bought with the precious blood of Christ
You'd be hard pressed to find much wrong with that, theologically speaking. But the way I've often heard it played in church the verse starts out quietly, with no instruments. Then at "Then bursting forth" the instruments all come in and, at one church I attended, you could predict when all the hands would go up pretty much to the second - it would be between "and" and "stands" of "and as he stands in victory". When you can predict with very high accuracy when all the hands will go up it's probably safe to say it's an emotional response rather than responding to the Holy Spirit, unless we want to argue that the Spirit moves in such formulaic ways it can be predicted so accurately.
There isn't necessarily anything wrong with an emotional response. If anything we should be concerned about the idea we can even remotely understand God's love for us without responding to it with some kind of emotion. The trouble is when songs are chosen to elicit an emotional response, when church consists of an hour of upbeat music followed by the suggestion (if not the outright statement) that the emotional euphoria people are experiencing is a mighty move of God, when people then try to recreate this "mighty move of God", and when people start to worry that God has apparently abandoned them yet again just as Monday morning starts and they wonder what they did wrong only to spend their time chasing the elusive high rather than seeking God.
If chasing the warm fuzzy feeling is all someone is about they might as well listen to "Ooh, Aah, Just a little bit" by Gina G and get much the same result. At least that way there's no pretense going on.