Why do Christians believe Christ's death applies to future sins? One of the points that different Christian denominations and movements generally agree on is that Christ died on the cross to atone for our sins, and his death was the ultimate sacrifice. This sacrifice can apply to anyone, both before and after his death.

My understanding of the sin offering and guilt offering in the Old Testament, which Jesus' death rendered no longer necessary, was that a person or nation committed an act, and subsequently a sacrifice was made to atone for that act. 

Why do Christians believe that Jesus' sacrifice is different to those of the Old Testament? That is, what is the Biblical basis for believing Jesus' sacrifice can apply to sins committed after his death?
My understanding of the sin offering and guilt offering which Jesus' death rendered no longer necessary was that a person or nation committed an act, and subsequently a sacrifice was made to atone for that act.

I'm going to focus on this bit in particular. It seems that you're suggesting that Jesus' death is a direct replacement for the Old Testament sacrifices, where one (past) sin was atoned for by one sacrifice - and therefore Jesus' sacrifice must work the same way. However, this the New Testament teaches that Jesus' death is completely different to this sacrifice system, and it not a direct replacement but God's plan from the beginning.
Why does Jesus' sacrifice work different the Old Testament sacrifices?
The apostle Paul, on several occasions, describes how Jesus starts of a new line of people who are saved, just as Adam was the first in a line of people who are not saved. Through faith, people are brought from one group into the other.

But the gift is not like the trespass. For if the many died by the trespass of the one man, how much more did God’s grace and the gift that came by the grace of the one man, Jesus Christ, overflow to the many! 16 Nor can the gift of God be compared with the result of one man’s sin: The judgment followed one sin and brought condemnation, but the gift followed many trespasses and brought justification. 17 For if, by the trespass of the one man, death reigned through that one man, how much more will those who receive God’s abundant provision of grace and of the gift of righteousness reign in life through the one man, Jesus Christ!
Romans 5:15-17 (NIV®)

Paul specifically highlights that Jesus death is able to cover over many sins. That Old Testament pattern of one sacrifice for one sin isn't required, because Jesus' death can cover many sins.
The writer to the Hebrews expands on this idea even further, claiming that the Old Testament sacrifices were merely a teaching aide, and didn't have any power to atone for sin at all.

The law is only a shadow of the good things that are coming—not the realities themselves. For this reason it can never, by the same sacrifices repeated endlessly year after year, make perfect those who draw near to worship. 2 Otherwise, would they not have stopped being offered? For the worshipers would have been cleansed once for all, and would no longer have felt guilty for their sins. 3 But those sacrifices are an annual reminder of sins. 4 It is impossible for the blood of bulls and goats to take away sins.
Hebrews 10:1-4 (NIV®)

However, when Jesus comes he does away with OT sacrifices, because he himself is the true sacrifice they pointed towards.

First [Jesus] said, “Sacrifices and offerings, burnt offerings and sin offerings you did not desire, nor were you pleased with them”—though they were offered in accordance with the law. 9 Then he said, “Here I am, I have come to do your will.” He sets aside the first to establish the second. 10 And by that will, we have been made holy through the sacrifice of the body of Jesus Christ once for all.
Hebrews 10:8-10 (NIV®)

Instead Jesus sacrifice applies to all people at all times, if they have faith in him.

Day after day every priest stands and performs his religious duties; again and again he offers the same sacrifices, which can never take away sins. 12 But when this priest had offered for all time one sacrifice for sins, he sat down at the right hand of God, 13 and since that time he waits for his enemies to be made his footstool. 14 For by one sacrifice he has made perfect forever those who are being made holy.
Hebrew 10:11-14 (NIV®)

And

And where these have been forgiven, sacrifice for sin is no longer necessary.
Hebrews 10:18 (NIV®)