The Cessationist-Continuationist debate is not one I enter into often. You can find far too many straw man arguments. And personal attacks. Cooler heads rarely prevail. It is not really a position you can “proof-text” and it polarizes people.
People often have a hard time distinguishing the ordinary from extraordinary. This distinction is made in the Westminster Confession of Faith with regard to means God uses to bring someone to saving faith (XIV, 1). For instance, should the ordinary means of hearing the gospel not be available, God may use extraordinary means to convert a person. Those cases are rare, and are not to be expected by us.
To lay my cards on the table, I am most fully persuaded by the cessationist argument presented by Richard Gaffin in Perspectives on Pentecost. I say “most fully persuaded” because I still do not experience what John Frame calls “cognitive rest” on the issue. Wayne Grudem’s book The Gift of Prophecy in the New Testament and Today is a worthy challenger. What he argues for sounds an awful lot like illumination. Where cessationists get unduly bent out of shape, using extreme comments, is acting like what one thinks is God’s word (notice, I didn’t use a capital ‘w’) to a person is universally binding.
A personal anecdote from before my ascent (or descent depending on your perspective) to Presbyterianism. I would regularly pray with a group of people including some charismatics. From my understanding of Scripture even then, the gift of tongues was not being used biblically (which is why I stopped attending). I pretty much knew what one guy would say in “tongues”, as he repeated the same verbal patterns each week. But that is not my point.
One week there was a guy there I’d never met before and never saw again. I was a recent college graduate looking for a job. I prayed about that. While another friend was praying about something the word ‘repent’ entered my mind but I didn’t think too much of it. After we were done, the new guy said that while I was waiting on God, He was waiting on me to be more proactive in my job search. He was right, I was being passive. But how could this guy know? I felt emboldened and shared the need to repent with my friend. I had no idea of what he needed to repent of. But it certainly struck a chord with him.
Being wrong on this doesn't make him a heretic, dudes.
In both instances, God brought something to mind for a particular person. In both cases the person had no idea of the circumstances of the other. I’m not sure what to do with that. But that is not a regular thing in my experience. Perhaps we Reformed non-charismatics just call it intuition. But I suspect that the Spirit is at work in intuition. Sometimes intuition is just our own creation, projecting past experience on something. So I am not prone to flaming Mark Driscoll over utilizing this in counseling situations as some have. Spurgeon spoke about personal circumstances of which he had no personal knowledge from the pulpit at times. I hold my cessationism lightly, not tightly.
Back to a possible argument for the cessation of the gifts. I’m currently reading through Numbers. Something in chapter 27 stood out to me.
18So the LORD said to Moses, “Take Joshua the son of Nun, a man in whom is the Spirit, and lay your hand on him. 19Make him stand before Eleazar the priest and all the congregation, and you shall commission him in their sight. 20You shall invest him with some of your authority. that all the congregation of the people of Israel may obey. 21And he shall stand before Eleazar the priest, who shall inquire for him by the judgment of the Urim before the LORD.
We see a shift here similar to that argued for by cessationists. Moses spoke to God face to face, like a friend. As the leader of the exodus generation he had an extraordinary spiritual life which was necessary for the massive shift in redemptive history.
He’s about to die and needs to commission Joshua as his successor. Note the continuity: Joshua has the Spirit. Note the discontinuity: he does not meet God face to face. He’s dependent on the ordinary means of casting lots. Something ceased! God did not gift all subsequent leaders of God’s people as He had Moses.
This is essentially the argument used by Gaffin and other Reformed cessationists. This is quite better than John MacArthur’s argument in my opinion. The Apostles, like Moses, had extraordinary gifts of the Spirit due to the radical shift in redemptive history. Yes, some other people had some extraordinary gifts to testify to the reality of this new message. But, while the subsequent generations of Christians had the Spirit they did not experience these extraordinary gifts or manifestations of the Spirit.
The biblical-theological argument, meaning there is no proof text but a case is inferred from the course of redemptive history, finds basis in Numbers 27. It functions as something of a precedent. This is why, unlike Mark Driscoll, I am not a charismatic with a seat belt. I don’t think tongues is a gift for today. While God may heal people, no one has the gift of healing (sorry Benny). I do think God brings Scripture to mind at critical moments in our lives to address our concerns or those of others. I do think God gives us insight into other people. But neither of these situations creates a normative situation for all Christians. Any such insight can be verified if about a person’s history, and any instruction must be measured by Scripture. Like Paul said to the Corinthians.
Cavman Considers 
So do cessationists tend to lean more towards dispensationalists?
No, dispensationalists are split on the issue.