For example, two melodies at once but one melody is only on the on beats while the other is only on the off beats, does this remove the harmony from counterpoint?
It does not make sense to try and remove the harmony from counterpoint as by definition it is the relationship between voices that are interdependent harmonically yet independent in rhythm and contour. Removing harmony defeats the purpose of counterpoint to make lines work together yet sound independent.
Doing what you suggested does not remove the harmony aspect of counterpoint and could even be considered fourth species counterpoint where you use a lot of resolving nonharmonic tones like suspensions.
Short answer: no.
Long answer: no. Harmony deals with notes presented simultaneously, and the succession of these simultaneities; counterpoint deals with lines presented simultaneously. Harmony in its broadest sense encompasses what we call tonal and functional harmony, but isn't limited to these.
The difference is one of viewpoint: harmony can be seen as a succession of vertical tranches of a polyphonic piece of music, whereas counterpoint views the music as a combination of horizontal layers. The concepts are orthogonal, and by no means mutually exclusive: a piece of polyphonic music may emphasise one or the other viewpoint, but the unemphasised dimension is always there as well.
But, even more specifically with regards to your question, the presence of syncopation isn't sufficient to derail functional harmony. Used between simultaneous tonal lines, syncopation will tend to suggest anticipations and/or suspensions. In common practice harmony, these are called non-harmonic notes, which are used to make melodic connections between harmonic tones. An anticipation states a tone that is foreign to the harmony on an offbeat, and holds it over to the downbeat where it forms part of the harmony, i.e., it resolves. A suspension works the other way around: the held note is part of the harmony on the offbeat, and foreign to the harmony on the following downbeat (usually resolved by moving by step to a chord tone on the next beat). Chains of suspensions are actually something of a cliché in tonal music:
This is from J. J. Froberger's 2nd Ricercare. It is a contrapuntal piece of music that is filled with precisely the kinds of syncopations that you mention; it is also a tonal piece, even somewhat modal (coming at the very end of the transition between modality and common practice tonality). The key is G with a strong Mixolydian cast. Let's have a look at where he sets the syncopations in relief, with the entrance of the descant voice on a high D at the end of m.75.
If you take a vertical slice through the 4th beat of m.75, you have a first inversion D minor chord (v6 in the context of G). The D is held over the first beat of the next bar, where it is dissonant to the E underneath, resolving to C on the 2nd crochet: the harmony for the first half of m.76 is thus the first inversion of a C major triad (IV6 in G).
The high C is held over the third beat of m.76 to form vii° with the F♯ and A beneath it. (Actually, the E left hanging in the alto gives a strong suggestion of a half-diminished seventh on F♯.) Movement by step produces a G major chord (I) on the last beat of the bar, the B of which forms a dissonance with C at the start of m.77, resolving to A as part of ii6 on the second beat. The third beat of m.77 articulates the third inversion of D7 leading to a first inversion G chord (I6) on the last beat.
In m.78, the held G in the descant is a common tone; the B in the alto isn't dissonant, but I would tend to call it an accented passing tone (as is the quaver F in the descant): the first half of the bar articulates a first inversion C major chord. The C in the alto is then held to form a minor seventh chord on D, the C resolving to B to form a first inversion diminished chord on B taken a vii° of the key of C, forming a cadence with the C major chord that starts the last bar of the example. (The quaver G in the tenor on the last beat of m.78 forms a 2nd inversion dominant 7th in the key of C, but Froberger treats it as an accented passing note.) Froberger modulates from G to C starting in m.78.
So there you have it: a passage in which at least one voice is syncopated against the rest at any given point of time, which also has a very strong harmonic framework. (Froberger was known for his harmony, as a matter of fact.) If you were to reduce the number of voices to two (which Froberger starts to do briefly at the end of the passage), it wouldn't change the fact the music implies harmonies and harmonic motion.
Now, it is entirely possible to remove the sense of resolution and functional harmony from a pair of voices syncopated relative to each other, but they will still tend to imply harmonic roots and a (perhaps inadvertent) succession of these roots. Syncopation is a rhythmic technique: it is quite neutral with regards to the presence or absence of functional harmony, which is established by voice leading.
two melodies at once but one melody is only on the on beats while the other is only on the off beats, does this remove the harmony from counterpoint?
No. The ear makes the alternating notes blend in our perception, and this creates the effect of a harmonic progression. Extreme case of this: an arpeggio! The notes of the chord are played, but not at the same time. Still, even though there's no chord, strictly speaking, we still hear the harmony, and we can think of the arpeggio as though it were a chord.