The major argument in favor of doing this is the idea that it would spread the power to countries that are just as deserving as some of the countries that are now permanent members.
The permanent membership of the Security Council is something of a relic of WWII. It includes only the countries who were on the side of the US in WWII. This ignores present day realities. It is hard to argue, for example, that France and Britain deserve more of a say than Germany and Japan on the Security Council.
The main argument against, in my opinion, is that it would give more countries the ability to veto actions, leading to even less credibility for the UN.
The United Nations was formed in 1945, immediately following World War II in hopes of avoiding future such wars. The victors, the Allied powers, each received a seat on the Permanent Security Council and an important perk that came with it: veto power.
In the modern day, Japan and Germany are democracies and major trading partners with us and the rest of the world, so they can make a credible claim that the world's second largest economy (Japan) and largest economy in Europe (Germany) deserve a seat at the same table.
The problem is, since the current members have veto power, they can simply veto any proposal to change the current makeup of the Security Council. Russia doesn't particularly like Germany and China does not particularly like Japan, and so their membership has always been vetoed or threatened with veto. Why should they vote yes when it diminishes their own power and influence?
Another reason it could cause problems is that with two new vetoes on board, it would be even more difficult for the UN to get anything done, when it's already pretty difficult now.