How do I change matplotlib's subplot projection of an existing axis?

You can't change the projection of an existing axes, the reason is given below. However the solution to your underlying problem is simply to use the subplot_kw argument to plt.subplots() described in the matplotlib documentation here. For example, if you wanted all your subplots to have the cartopy.crs.PlateCarree projection you could do

import matplotlib.pyplot as plt
import cartopy.crs as ccrs

# Create a grid of plots
fig, (ax1, ax2) = plt.subplots(ncols=2, subplot_kw={'projection': ccrs.PlateCarree()})

Regarding the actual question, specifying a projection when you create an axes set determines the axes class you get, which is different for each projection type. For example

import matplotlib.pyplot as plt
import cartopy.crs as ccrs

ax1 = plt.subplot(311)
ax2 = plt.subplot(312, projection='polar')
ax3 = plt.subplot(313, projection=ccrs.PlateCarree())

print(type(ax1))
print(type(ax2))
print(type(ax3))

This code will print the following

<class 'matplotlib.axes._subplots.AxesSubplot'>
<class 'matplotlib.axes._subplots.PolarAxesSubplot'>
<class 'cartopy.mpl.geoaxes.GeoAxesSubplot'>

Notice how each axes is actually an instance of a different class.

Assuming there are multiple axes being used for 2D plotting, like...

fig = matplotlib.pyplot.Figure()
axs = fig.subplots(3, 4) # prepare for multiple subplots
# (some plotting here)
axs[0,0].plot([1,2,3])

... one can simply destroy one of them and replace it with a new one having the 3D projection:

axs[2,3].remove()
ax = fig.add_subplot(3, 4, 12, projection='3d')
ax.plot_surface(...)

Just note that unlike rest of Python, the add_subplot uses row-column indexing starting from 1 (not from 0).

EDIT: Changed my typo about indexing.