Actually a current transformer won't work -- it only handles changing current, not DC. There are some similar Hall-effect sensors which do handle DC, but they are expensive.
A resistor should not generate 16 V spikes across it (even if the motor does generate large spikes), and it is strange that the DVM even picks those up.
1) Is the resistor in the ground line of the motor ? Connecting it in the supply line is possible, but increases the complexity (and errors) of the opamp circuit.
2) I would assume that the R is low enough in value that 'normal' currents only generate a few 100 mV across it -- if so, then a normal diode connected in the forward direction would clamp the voltage to < 0.6 V.
3) Be sure that the 'top' end of the R goes to the op amp circuit's oinput (with a separate wire if necessary directly from the R), and the bottom end of the R goes directly to the opamp's ground. This is called a 'Kelvin connection' and avoids errors which effectively make it appear that the R is a different value than you expect.
4) Use an opamp in a non-inverting configuration (senseR -> '+' input; R1 from output to '-' input; R2 from '-' input to GND. Wire the opamp GND, bottom of R2, and Teensy GND together. If your sense R dropped (say) 100 mV, then a gain of ~ 10x may be appropriate -- make R1 9k, and R2 1k.
5) Choose an opamp that has a common-mode range that includes ground (0 V). Most new ones do.
4.5) power the opamp from the Teensy's 3.3 V supply -- that will avoid overloading the analog inputs.
6) Be aware that when heavily loaded (or stalled, or just starting up), a motor could draw 10x or more current than when lightly loaded. If you need to track that, then you may not be able to use the values above.
7) Given that motor behavior is not particularly consistent, and you probably don't need incredible accuracy. Consider eliminating the opamp and just using the inputs from the sense R directly and configure the ADC to use the PGA (internal preamp) suitably. To avoid serious mishaps, do consider using a protection R (1k..10k) from the sense R (which is probably in the 1 ohm range ?) to the ADC input.