I'll start by saying I oppose ANWR drilling and try to explain why as well as address the arguments put forth by those posters in favor of drilling.
"after the oil is gone, so is the hardware" - Oil extraction does not just leave behind a few rusty bolts and concrete slabs, there are environmental costs in the extraction as well. http://pubs.usgs.gov/wri/wri03-4260/
"we need to drill so we pay less at the pump" - The US Energy Information Administration put forth this report http://www.eia.doe.gov/oiaf/servicerpt/anwr/introduction.html that clearly states that at most ANWR drilling would move prices by about $1.44/barrel, which if you read the full report says that this savings could easily be counteracted by OPEC.
"oil companies can extract oil with little to no environmental impact" - They have been getting better at this, but I hope no one has forgotten the Exxon Valdez oil spill. Windmills and solar panels can't fall off a truck and pollute 1300 miles of coastline
"Studies show that humans are not to blame for global warming, we are actually in a natural cooling trend" - Global warming does not mean that everywhere gets hotter, some places get colder, the important number is average global temperature, a more correct term would be global climate change. I would also argue that climate change can be ignored in this oil argument. I believe this is a moral issue, are we good stewards of our planet. "99.9999% of us are NEVER going to visit the roughly .0001% of the nearly impossible to get to anyway frozen arctic location where the drilling would take place". Just because I spend 99% of my life in my town/city does not make it ok for me to send my trash to your state. I don't visit Yellowstone everyday, but I don't want it turned into a strip mall either.
"Most automobiles produced are limited to 0 emissions during inspections. CO2 is the largest complaint for fossil fuel emissions today. Yet it is also the most necessary compound for the continuance of plants and trees which emit the O2 that you are so worried about." - If our cars are limited to 0 emissions then you should have no problem sucking on your tailpipe. Earth does not need us to help produce CO2, it does that just fine on its own through volcanoes, organic matter decay etc. which has been in balance for the last several million years, the difference is that the CO2 a decaying tree emits is offset by the oxygen it produces, however burning fossil fuels releases CO2, not to mention heavy metals, which previously were sequestered harmlessly under the ground, but are now free to enter our food and water supply. It is no longer a balanced system.
"U.S./World economy CANNOT continue to handle energy prices that are continuing to sky rocket" - So do something about it, I replaced all of my traditional light bulbs with compact fluorescents and saw my power bill drop by $10 a month or $120 a year and that was for a small 2 bedroom house (about a 1000 square feet). I have my electric water heater on a timer so that it only turns on to give hot water in the morning and at night, this saves me on the order of $200 a year or more. Insulating your home makes a big difference as well. In one old house I rented I talked the landlord into letting me insulate the attic. I spent about $200 on insulation (it was a pretty small house) and saw the gas bill in winter drop from $130 a month to $75 a month. Don't look to the government to provide you with cheap energy so you don't feel the financial cost of wasting it. Help yourself.
"While it is tragic that we have lost 4,000 soldiers in the War on Terror, this loss is small in comparison to the number Americans that sacrified their lives in Vietnam or the number of people of that die on our highways every year" This speaks to Americas self entitlement that somehow our lives are more valuable than those of other countries. I don't like to rely on Wikipedia, but there are enough links to reliable sources here, http://en.wikipedia.org/wiki/Casualties_of_the_Iraq_War ,to make the argument that the cost has been much higher than 4000 American soldiers (by some estimates over 1,000,000 Iraqi people), unless of course you think one American life is worth that of 250 or so Iraqis.
"windmills and solar panels are and eyesore" - This is really an opinion and emotional argument, one not based on logic. I happen to find wind farms beautiful, in fact I look forward to driving past one particular wind farm in Texas on my way home as it breaks the monotony of the flat countryside. Offshore wind farms are also another solution, no one has it in their back yard and the natural winds created at the coast make them better producers. You also don't have to fill a field with solar panels, how about a few of these on your house, http://www.energyrefuge.com/archives/solar_shingles.htm They are barely noticeable or could be put on a side not facing the street, if every house had a few of these power cost/production needs would drop greatly. That's also the nice thing about solar panels, they are most efficient when you need them most, on a hot sunny day when you have your AC cranked down to 68 degrees. Although if you don't care about ANWR, which "99.9999% of us are NEVER going to visit" then why should you care about a solar farm in the middle of the desert which I also doubt you are going to visit.
If you made it to the end, thank you for taking the time to read about an opinion that may differ from yours. That is the key to solving energy problems, education and debate, rather than throwing around 10 second sound bites you heard on the news as arguments, take the time to research and draw your own conclusions. Finding more oil is not a solution, as India, China, Africa etc. continue to build their economies, they use increasingly more energy and they outnumber us in population by a significant amount. This increased competition makes alternative sources of energy the only real solution. I know we won't stop using oil tomorrow, or perhaps ever, but I believe we can move forward. I'm sure everyone was happy when they no longer had to use candles at night to see, how happy would you be if your house produced enough energy through solar/wind power that you no longer received a power bill?