Say the fibre is for 700 nm, and we shine light at 1064 nm into it.
Would it change the mode? The polarisation? The beam shape (i.e. introducing ellipticity in the beam)?
The manufacturer generally specifies a cutoff wavelength for single mode optical fibre designed for specific wavelength. For example, a single mode fibre desgined for 700nm would have a cutoff wavelength few tens of nanometers below 700, lets say 680nm. This means that for wavelengths longer than 680nm, the Normalized Frequency or the V-number of the fibre would be equal to or smaller than 2.405, which is a condition for single mode propagation. If the V-number is higher than 2.405 then multiple modes are excited.
Because you are using the single mode fibre at higher wavelength (1064nm) than it is designed for (700nm), the V-number is still lower than 2.405 and you would get single mode propagation. However, the optical power would be attenuated more than the specified value at 700nm. This is because, at 1064nm the mode field diameter is larger than at 700nm, and slightly higher fraction of the beam propagates through the cladding material.