The driven element of a yagi antenna is normally a half wave dipole, and it's true that a dipole antenna is resonant on multiple harmonically related frequencies. An ideal 10 m long dipole for example is resonant on the frequencies where its length is an integer multiple of one half wave length for each frequency ie: 15 MHz (.5 λ), 30 MHz (1 λ), 45 MHz (1.5 λ), 60 MHz (2 λ), etc.
However the distances between the elements along the boom of a yagi required to provide forward gain and reverse rejection must be matched exactly to suit the wavelength of the frequency of operation, and unfortunately these required distances aren't harmonically related in the same direct way as for element length and dipole resonance and don't follow the same rules.
To explain in more detail, the distances between the elements and the element lengths are precisely chosen so that RF energy induced in the elements from re-radiation from the other elements adds in the forward direction, and subtracts in the reverse direction, which gives the antenna forward gain and a front to back ratio. The interaction between element lengths and positioning is complex and only works at the design frequency of the antenna.
Another problem is that the impedance of the antenna will only be correct at the design frequency. At 14 MHz, if the impedance is perfect at 50 ohms, then the SWR will be good. But when operation is changed to 28 MHz, the antenna impedance will change and the SWR will be bad.
To elaborate further, for a dipole in free space, the fundamental resonant frequency is determined purely and only by the impedance of free space and the length and cross-sectional area of the antenna elements, and at resonance, the feed point will present a resistive impedance with zero reactance. If the dipole is an odd integer multiple of a half wave length (in your case it's 1 x half wavelengths at 14 MHz) the resistive impedance will be between about 70 Ω and a few hundred ohms, but if it's an even integer multiple of a half wave length (it's 2 x half wavelengths at 28 MHz), then the impedance will a very high value which is difficult to work with. Incidentally the exact value of impedance is related to the number of wavelengths and is independent of the actual frequency.
Having said that, without a matching network, the feed point impedance for a yagi at resonance is much less than 50 ohms, due to the presence of the reflector and director elements. To compensate for this, the length of the driven element is often chosen to be slightly less than a half wave length so that a matching network such as a gamma match can be used to cancel out the resultant inductive reactance caused by the shortened dipole while at the same time allowing an exact match to 50 ohms to be obtained. This arrangement is highly sensitive to the frequency and a correct match at 14 MHz definitely won't be correct at 28 MHz.
To complicate matters, the presence of the additional elements which surround the dipole driven element have an effect on it's electrical length, and this effect is frequency dependent. So if the antenna is tuned say for 14.000 MHz, if you double the frequency to 28.000 MHz, the antenna dimensions haven't changed, and the effect of the nearby reflector and director on the electrical length of the driven element won't be the same at the new frequency, and the impedance will change for that reason also.
So to answer your question, a 3 element yagi designed to work with good SWR, gain and front to back ratio on 14 MHZ, when used on 28 MHZ, won't behave like a yagi and will have little gain or front to back ratio, and the impedance will change and the SWR will be bad.
Hope that helps !