Why [FeF6]3– is colourless whereas [CoF6]3– is coloured ?

[FeF₆]^3- is a Fe (III) complex hence [Ar] 3d⁵. F^- is a weak field ligand and the complex is high spin: 

                                     t₂g (↑) (↑)(↑)   →Δ_oct→     (↑)(↑)eg.

And there is 5 unpaired electrons hence it's spin multiplicity = (2S+1) = 6  , and multiplicity of  a excited is cannot be six thus The transitions in Fe ⁺³ ion are spin forbidden and are extremely weak so as to make [FeF₆]^3- almost colorless.

[CoF₆] ^3- this is a Co (III) complex [Ar] 3d⁶. Again F^- is a weak field ligand and [CoF₆]^3- is one of only two common cmplxs that is high spin:

                       t₂g (↑↓) (↑)(↑)     →Δ_oct→      (↑)(↑) eg

Only one band is expected namely the t₂g→eg transition (d¹). The complex is blue consistent with this transition being in the low energy red. 

Why [Mn(H2O)6]+2 is colourless although in which Mn+2 ion had five unpaired electrons ?

There are 5 unpaired electrons in Mn⁺² ion and we know that.
                                 Spin multiplicity is = (2S+1) 
                     Where S is some of spin thus  S = 5×1/2=5/2
                                            Hence multiplicity = (2S+1)= 6 

we know multiplicity of excited state cannot be six, thus electronic transition in Mn⁺²  are spin forbidden, hence [Mn(H₂O)₆]⁺² salt appear colourless.

Also [Mn(H₂O)₆]⁺² has centre of symmetry and in such cases electronic transition are expected to be laporate forbidden.

Why Fe(CO)5 is colourless while Fe(bipy)(CO)3 is intensely purple in colour ?

The intense colour of the latter complex is strongly suggestive of a charge transfer transition and since the metal is already fully reduced (zero oxidation state), it is highly likely that this involves a MLCT transition. The π* levels of the bipy or CO ligands are possible acceptors but the fact that Fe(CO)₅ doesn’t show this colour suggests that it is the bipy π* levels that are involved in Fe(bipy)(CO)₃. Since there should be MLCT transitions to the CO π* levels as well, we assume that the lack of colour for Fe(CO)₅ means that these transitions fall in the UV rather than the visible.