Add to lib/cartopy/io/img_tiles.py to enable loading image tiles from an MBTiles file. Unfortunately the demo (see comments below) does not currently work due to new bug: https://github.com/SciTools/cartopy/issues/312

img_tiles.py

class MBTiles(GoogleTiles):
    ''' Retrieve tile data from MBTiles sqlite database instead of a url '''
    def __init__(self, tile_db):
        self.tile_db = tile_db
        GoogleTiles.__init__(self)

    def get_image(self, tile):
        import cStringIO
        import sqlite3

        x, y, z = tile
        sql_select = ('select zoom_level, tile_column, tile_row, tile_data' +
            ' from tiles')
        sql_where = 'where zoom_level=%s and tile_column=%s and tile_row=%s;' \
            % (z, x, y)
        con = sqlite3.connect(self.tile_db)
        tiles = con.execute(sql_select + ' ' + sql_where)
        tile_from_db = tiles.fetchone()

        try:
            im_data = tile_from_db[3]
        except TypeError:
            raise TypeError('No image data available for requested tile')

        img = Image.open(cStringIO.StringIO(im_data))
        img = img.convert(self.desired_tile_form)
        return img, self.tileextent(tile), 'lower'

import matplotlib.pyplot as plt
import cartopy.crs as ccrs
from cartopy.io import img_tiles

ax = plt.axes(projection=ccrs.Mercator())
tiles = img_tiles.MBTiles('osmbright.mbtiles')
ax.add_image(tiles, 8)
plt.show()

Cartopy sends a Google tile index, but the database uses TMS indexing.

Easy fix: y = (1 << z) - y - 1