Faucet aerators are cheap ways to help save water and you can install/uninstall them pretty easily.
SmartStrips and other power-saving devices can help cut off phantom loads.
Most innovations that reduce plug load electric usage, except perhaps large appliances, are available to renters.
Energy-saving light bulbs can also generally be installed/uninstalled pretty easily.
In most cold-climate rentals, clear plastic window film can also be put over windows to help keep heat in in the wintertime. I've seen renters with chimneys put in an insulation roll and some plastic film seal to help prevent the heat from going up and out as much (do ask the landlord on this).
Sustainable living techniques that are related to food (e.g. reduce waste, consider sources), transportation (e.g. bike, take mass transit), and behavioral changes (e.g. shorter showers) are also available. See Zero Waste Home (book or blog) for a lot of ideas.
If recycling and/or compost are available, having easy-to-use recycling bins (and using them) is also a good idea.
Houseplants can improve your indoor air quality and if those are edibles (e.g. basil) you can get the benefits of eating fresh local herbs (for basil, pinch off the tops before any flowers develop).
If you pay the utility bills, the utility might have rebate programs for certain energy efficiency upgrades like a programmable thermostat. Something like that you would need the landlord's permission for but it might be a decent idea to ask.
As noted above, some energy-efficient upgrades are things you can easily install/uninstall, and you could keep the inefficient elements around to reinstall when you leave, OR when you're putting them in you could describe the benefits & ask the landlord if they want to keep that and are willing to apply your receipts from the purchases toward a discount on rent. Sometimes the "payback period" is even within the time period you'll be staying there, especially if you're considering environmental/health/livability/other noneconomic benefits.