Tax Algebra:

Let dx(Px,Py,I) and dy(Px,Py,I) be the demand for X and Y given prices and 
income.  These are observable, and they are by definition the most
preferred choices on the budget line 

(B1)            Px*X + Py*Y = I

Now impose a tax on X and give a rebate Z.  The new budget constraint is 

(B2)            (Px+T)*X + Py*Y = I + Z, 

and the new optimal choices are given by dx(Px+T,Py,I+Z) and 
dy(Px+T,Py,I+Z), which are as usual determined by finding the indifference 
curve which is tangent to the new budget constraint (B2).  

In principle, we want to know how much Z we have to give to make the 
consumer as well off as she was before the tax and rebate.  This requires 
knowledge of the utility function, however.  So instead, let's ask a related 
question: 

(Q)  If we gave a rebate to the consumer which has the property that the rebate
is equal to the amount the consumer pays in taxes, will the consumer be as well
off as without the tax and rebate?

The answer to this question tells us something about the distortionary effect 
of the tax, and the waste which it creates.

To answer the question, we first find the rebate Z where an optimizing consumer
pays Z in taxes.  That is, we find the particular value of Z so that the 
following equation holds:

(3)             T*dx(Px+T,Py,Z+I) = Z

At this (and only this) value of Z, the consumer's tax receipts exactly equal 
her rebate.  Now, is the consumer better or worse off?

Let us check the consumer's new budget constraint (B2) evaluated at the optimal
 choices given taxes:

   (Px + T) * dx(Px+T,Py,Z+I) + Py*dy(Px+T,Py,Z+I) = Z + I

Substituting (3) on the right-hand side and cancelling, we get 

   Px*dx(Px+T,Py,Z+I) + Py*dy(Px+T,Py,Z+I) = I

This equation tells us that the total expenditure on X and Y, when taxes and 
rebates are netted out, is I.  That is, (dx(Px+T,Py,Z+I),dy(Px+T,Py,Z+I)) lies 
on the budget line (B1).

Now, we have our result.  We know that (dx(Px,Py,I),dy(Px,Py,I)) is the most 
preferred point on the budget line (B1), by definition.  So, in particular, 
(dx(Px,Py,I),dy(Px,Py,I)) must be preferred to 
(dx(Px+T,Py,Z+I),dy(Px+T,Py,Z+I)), since the latter point is also on (B1).
[This is a "revealed preference" argument.]

If (dx(Px,Py,I),dy(Px,Py,I)) is optimal, why is it not chosen?  Well, recall 
that in the new situation, the consumer faces budget constraint (B2), not (B1).
(The consumer ignores her effect on Z when she makes her choices, that is, 
she treats Z as fixed).  

The choice dx(Px+T,Py,Z+I),dy(Px+T,Py,Z+I) *is* optimal, given the prices 
and income faced by the consumer (B2).  We have simply chosen Z so that B2 
and B1 cross exactly at (dx(Px+T,Py,Z+I),dy(Px+T,Py,Z+I)).  

In other words, if the consumer could spend all of I tax-free at the stores, 
instead of bothering with the government at all, the consumer could make a 
better choice, and achieve higher utility.  When the government is involved, 
the consumer still exhausts her income I but the best she can do given the 
government-interference-budget-constraint (B2) is consume the bundle 
(dx(Px+T,Py,Z+I),dy(Px+T,Py,Z+I)).

This same analysis holds for any set of consumer preferences -- as long as 
consumer behavior satisfies the weak axiom of revealed preference, government 
intervention necessarily creates a distortion.  We have proved that taxes cause
harm to the consumer even if the consumer receives enough rebate to push her 
back onto her old budget line.

I hope this clarifies things -- if you're still confused, feel free to come by
my office hours.