There is no contradiction because the two verses are teaching separate doctrines
It appears you believe the Isaiah verse is teaching that God will forgive all people regardless their actions while the 2 Cor. verse is teaching our actions matter. That's not quite right because you're comparing apples and oranges.
Let's start with your verse in Isaiah in context:
Thou hast bought me no sweet cane with money, neither hast thou filled me with the fat of thy sacrifices: but thou hast made me to serve with thy sins, thou hast wearied me with thine iniquities. I, even I, am he that blotteth out thy transgressions for mine own sake, and will not remember thy sins. Put me in remembrance: let us plead together: declare thou, that thou mayest be justified. (Isaiah 43:24-26)
Looking at the verses surrounding the Isaiah reference, we see that Jehovah is trying to make a point. "You've treated me badly, even I, the person with the power and authority to forgive. Instead you should be remembering and honoring Me."
From this verse we learn that God has the power to forgive, and when He does forgive, He remembers them no more.1
Now, before we jump to your verse in 2 Corinthians, let's look at a couple more verses. It's important to avoid comparing just two single verses out of context. It's so easy to come to an incomplete conclusion!
For he saith to Moses, I will have mercy on whom I will have mercy, and I will have compassion on whom I will have compassion. So then it is not of him that willeth, nor of him that runneth, but of God that sheweth mercy. (Rom. 9:15)
Now it's a bit clearer, but we can still jump to an incomplete conclusion. We know God has the power to forgive and choose never to remember. Now we learn that He will show mercy where He wishes and that He is not obligated simply because of the actions of people (not of [people] who willeth, nor of [people] that runneth, but of [the choice of God alone]). But this makes it sound like any choice we make to do good is irrelevant.
Let's chase down that reference Paul makes to Moses.
And he said unto him, If thy presence go not with me, carry us not up hence. For wherein shall it be known here that I and thy people have found grace in thy sight? is it not in that thou goest with us? so shall we be separated, I and thy people, from all the people that are upon the face of the earth.
And the Lord said unto Moses, I will do this thing also that thou hast spoken: for thou hast found grace in my sight, and I know thee by name.
And [Moses] said, I beseech thee, shew me thy glory.
And [the Lord] said, I will make all my goodness pass before thee, and I will proclaim the name of the Lord before thee; and will be gracious to whom I will be gracious, and will shew mercy on whom I will shew mercy.
And [the Lord continued], Thou canst not see my face: for there shall no man see me, and live.
And the Lord said, Behold, there is a place by me, and thou shalt stand upon a rock: And it shall come to pass, while my glory passeth by, that I will put thee in a clift of the rock, and will cover thee with my hand while I pass by: And I will take away mine hand, and thou shalt see my back parts: but my face shall not be seen. (Exodus 33:15-23)
There is a lot of great doctrine here, but the important part we need to take away I highlighted in bold. Though God is in no way obligated by anything we do (e.g., He can't be forced to do anything), He can and will show us mercy because of the things we've done.
- God has the power and authority to forgive (show mercy).
- God is in no way obligated by anything we do, it is always His choice.
- God may choose to honor us with mercy due to our good works and has done so in the past.
Which brings us to your second verse.
For we must all appear before the judgment seat of Christ; that every one may receive the things done in his body, according to that he hath done, whether it be good or bad. (2 Cor. 5:10)
Which now makes sense when compared to Isaiah.
- And therefore we choose while here on Earth to follow the Way of Christ, because we seek the mercy of God, and knowing that He cannot be compelled to give us that mercy, we act to gain His favor (to find grace in His sight), that He might show us mercy as He did Moses.
Therefore there is no contradiction because the two verses are teaching separate doctrines. The verse in Isaiah teaching that God has the power and authority to forgive and the verse in 2 Cor. teaching that when it comes to obtaining that forgiveness, our good works are a good start.2
1 This has always impressed me. Being an omnipotent God, He obviously has the ability to remember our sins whenever He wants. But, unlike mortals who conveniently remember the transgressions of others whenever it suits them, He can be trusted to never do so for our sake so long as He makes the choice to forgive. Cool! Now it's my job to secure that forgiveness, and I'm required to do it His way, not mine.
2 This discussion about keeping things in context is so very important, otherwise people will read Heb 8:12 or Heb 10:17 and jump to the conclusion the OP has that all sin is simply forgiven. In Hebrews, Paul is making an argument to the Jews that the Mosaic covenant isn't enough and that it has been replaced by the new Christian covenant. Those two verses are surrounded by discussions of new laws, new commandments, new expectations for living our lives. They are not simple, blind declarations that regardless the kind of life we choose to live, we'll be saved. (If you think about it, that obligates God, which will never be true.)