If Protestants won't ask dead saints to intercede because there is no mediator but Jesus then why do they ask living Christians to pray for them? Protestants reject the Catholic and Orthodox practice of asking deceased saints to pray for them, and I think there are two main reasons why they reject it: it is communicating with the dead, and it is dishonouring to Jesus. The first has been dealt with in many other questions, but I'd like to focus on the second one to see how valid an argument it is.

So the second major reasons why protestants don't ask the deceased to pray for them is that they believe they need no other mediator than Jesus. For protestants, asking anyone other than Jesus to intercede with God for them is dishonouring to Jesus!

But Protestant Christians have no hesitation at all in asking living Christians to pray for them. Why does the argument against dead Christians not apply to living Christians? If you ask your Christian brother or sister to pray for you, why is that not seeking another mediator than Jesus? The Bible clearly instructs us to pray for others, and it frequently show that early Christians asked other Christians to pray for them (ex. in Paul's letters), but does it ever instruct us to ask others to pray for us?
Intercession is not the same as mediation!

As other answers have said perfectly adequately, there is plenty of biblical support for interceding for one another and the example of the Apostle Paul requesting this intercession from other (living!) saints; although there is no particular scriptural warrant to explicitly endorse asking for such intercession from saints who have 'fallen asleep'.

So how is mediation different? Mediation implies there is a separation between parties - one side is estranged or in conflict with the other and does not have the capacity in itself to resolve the situation: this is descriptive of the situation prior to regeneration - our sins separate us from God, we can do nothing to resolve this situation of ourselves; when we receive Christ however, we are reconciled to God - no longer separated. Only Christ (not Mary or any of the other saints) could and can do this.

Although Intercession can be mistaken for mediation because it is sometimes (and biblically) referred to as 'standing in the gap', it does not actually require there to be disagreement between parties; in fact effective intercession relies on agreement between the will of pray-er and God and to a lesser but still real extent, the willingness of the object to receive the answer to prayer. Paul was not far from the Lord when he requested the prayers of the saints on his behalf, he merely sought that they join together with him in prayer that all might be agreed for God's will to be done in Paul's life and to share in the joy of seeing those prayers answered. When we intercede for 'sinners' (most commonly, unsolicited), the aspects of it involving mediation (ie addressing 'the gap') necessarily rely on Christ's work as the mediator, we don't (can't!) do that particular part of the work of intercession.