The Bible does indeed command us to fear God, in many places (Ecc. 12:13 and 1 Peter 2:17). It also tells us we are to love God (Matt. 22:37). Moreover, the biblical record does say "perfect love casts out fear" in 1 John 4:18. However, this supposed contradiction is demonstrably fallacious, on two counts.
First, there is the issue of the semantic range fallacy. This fallacy states that some word takes on each and every meaning of its possible usages each time it is used. A deviant of this fallacy applies here, where nearly the opposite takes place: it assumes a univocal usage for the word. That is, every time the word appears, it is assumed to be infused with the same meaning in every case. This is demonstrably fallacious. So what the objector would need here would be a reason to think "fear"is in the exact same sense in all uses.
Second, there is the issue of equivocation. That fallacy is making an argument whose terms appear to carry the same meaning, but in reality do not (and hence are different terms after all). This can be seen from the context of each verse. Fearing God, in the context of the injunction for believers to do so, very clearly means something like being in awe, admiration, and subjugation. Fear, in the more common and modern context (as well as the context of there being no fear in love) clearly means terror in judgment (v. 17). But Christians do not have to worry about the terror of judgment; because Christ died for our sins, we may go to that judgment with boldness. A simple reading of the chapter would clear that up.
Now it occurs to me the saying was probably not meant as an argument, but rather as a joke of sorts. But people that tend to make these jokes typically do so out of a place of truth. That is, they probably believe the Bible is a silly book, hopelessly mired in contradiction. This article has shown that at least this so-called contradiction is easily resolved.