not lagging behind in diligence, fervent in spirit, serving the Lord;
The main interpretive issue in this verse is the identification of the spirit, whether it should be understood as the Holy Spirit (cf. Moo, p. 778) or the human spirit (cf. Murray, 2:130; Hodge, p. 396; Harrison, p. 133). Morris (PNTC) and Schreiner (BECNT) both take a middling sort of view that both the Holy Spirit and the human spirit are in view.
Schreiner says, "Believers are to burn and seethe in their spirits, but the means by which this is done is the power of the Holy Spirit" (p. 665). This is true enough, but is it what Paul was saying? Was Paul intending some sort of double reference here with the word Spirit?
The "both/and" sort of view is a fairly common hermeneutical bailout that, in my estimation, is usually illegitimate.* One of the fundamental truths about language is that words can only mean one thing in a given context. The ability to communicate cannot bear up under the weight of the full semantic range. This is what James Barr (and later Carson and others) called "illegitimate totality transfer." It essentially means attaching the full semantic range as the meaning of an individual use of a word.
This is seem by some pastors and teachers who, seeming desperate to say something profound about a passage, seem almost to preach right out of the Greek Lexicon, waxing eloquent about the various meanings of a given word rather than honing in on the one particular meaning in the context at hand. This is both time consuming as well as non-productive. If the word in question means the same thing in another passage as it does in this one, then comparison may help to inform and fill out our understanding. If it means something else, comparison probably serves only to confuse. As a pastor, I need not confuse my hearers by appealing to passages that are irrelevant because the same word there means something different.
Back to Romans 12:11, the word for spirit is a word that can be used both for the human spirit and the Holy Spirit. But it cannot be used in both senses in the same usage. It means one or the other.
In Romans 12:11, the meaning is found in comparing the first part (don't be lazy) and the second part (be fervent). We would be ill-advised to press the parallelism to determine the similarities or differences between "diligence" and "spirit." But we would be well-advised to note that the opposite of lazy is fervency, not the Holy Spirit. So Paul was exhorting these living sacrifices (v. 1) whose lives are being transformed by the renewing of their minds (v. 2) to not be lazy but rather to be energetic in the service of the Lord.
*I suppose, if words mean anything, that I have just accused Schreiner and Morris, both highly respected NT scholars, of this hermeneutical bailout. But what else do we say? Whatever pneumati (the Greek word for Spirit) might mean in that passage, I cannot for the life of me figure out how it can mean both the human spirit and the Holy Spirit.