Why doesn't a transmission line mess up an antenna resonance? I don't understand why the resonance of an antenna is not affected by the transmission line to the receiver. For example, given a dipole antenna like the following:

The antenna length is carefully chosen to create a resonance at the target frequency, however, isn't this ruined by the transmission line essentially lengthening the effective length of the antenna?
 A: A dipole is fed by a frequency varying voltage [or current] source at the center between the two halves. In theory, you send a sinusoidal signal down the transmission line which will see the dipole two dipole leads as an impedance $Z_{ant}$. For a half-wave dipole, $Z_{ant} \approx 72 + j42.5$ $\Omega$.
The classical textbook analysis of the radiation from a dipole begins by ignoring the feed and assuming the sinusoidal voltage source as above to calculate the theoretical radiating fields and radiation resistance $R_{rad}$. Then, texts will introduce the concept of balanced impedance lines and explain that a coaxial feed is an unbalanced line, i.e. current flows on the surface of center conductor. (Recall that the modal propagation down the line is a TEM wave). Since a dipole requires a balanced feed, you can use a balun to change the unbalanced line (uneven currents) to balanced line (with equal currents) exciting both halves of the dipole. This also serves the purpose of "choking" (or blocking/grounding) any currents that could be induced down the shield of the coaxial feed.
Textbooks often do not discuss the impact of the feed structure itself on the radiation. Yes, this metallic line will scatter fields incident on it, and thus interact with the radiation pattern. When you measure the gain in an anechoic chamber, you should see this variation in the measured omni pattern and change front-to-back ratio (if you define such a thing for a dipole...). However, It shouldn't be much since the line is usually assumed to be orthogonal to the polarization of the radiating fields.
To answer your question directly, the transmission line doesn't lengthen the dipole feed. Like I said above, the fields propagating down the line are TEM, so basically it's just a voltage source exciting the dipole leads; with a properly attached balun, you will excite the half-wave sinusoidal current distribution on the dipole.
