Placing plants such as a spider plant into a bird cage would not harm the bird, and it is more likely that the bird will harm it. Leaves on the spider plant that may have been affected by the bird can easily be fixed if you trim it regularly. Spider plants are inexpensive and easy to look after so that is a great choice of plant to place in the cage with the bird.
However, be careful if you were to use pesticides on the plant as some pesticides can be toxic to birds. These include systemic pesticides that are often mixed with water or soil and take up the plant's structure, in turn making the plant poisonous to insects and most likely birds.
Before purchasing the plant, ask to make sure that the pesticides used on the plant beforehand are safe. If you aren't 100% sure they are, rinse the foliage and re-pot the plant. If you are still concerned, keep the plant away from your bird and let the plant grow for a few months before placing it in there with it.