As pointed out by G. Martens doping may have some impact on the crystal lattice. The main question is, however, what is the reason of the doping? You can have doping which does not affect the crystal lattice at all (in solid solutions, i.e. Au-Ag).On the other hand, you can strain the material to prevent phase transformation (e.g. stabilized ceramics). However, doping is perhaps more often used for controlling of electronic properties in semiconductors. It would be horrible if this would generate lattice defects which are negative for the live time of certain devices.

Could you tell us which kind of doping you are talking about? 

In doped material some other kind atoms are introduced into the host material in different ways. But in general the volume of the dopand  atoms does not fit into the lattice structure of the host material. Therefore the host lattice is locally distorted giving rise to various kinds of lattice defects in the worst cases.

As pointed out by G. Martens doping may have some impact on the crystal lattice. The main question is, however, what is the reason of the doping? You can have doping which does not affect the crystal lattice at all (in solid solutions, i.e. Au-Ag).On the other hand, you can strain the material to prevent phase transformation (e.g. stabilized ceramics). However, doping is perhaps more often used for controlling of electronic properties in semiconductors. It would be horrible if this would generate lattice defects which are negative for the live time of certain devices.

Could you tell us which kind of doping you are talking about? 

The topic for decades of research :-) Knowing host material and the dopant can narow it down and then some reasonable answers can be expected...

In this discussion, the doping of semiconductors has been referred.  There are two types of doping i) lightly doping (< 1% weight of host materials) and ii) heavily doping. Changes occur in parent materials as per Hume-Rothery Rules, Formulated a set of rules for alloying to predict when the substantial solubility can be expected.  The two atoms must have

i) the difference of atomic radii should not exceed 15%,

ii) the difference of electro negativity (chemical affinity) should be small.

iii) the number of valence electron should not be very different

iv) identical crystal structure

Not obeying this rule, lattice mismatch occurs.  Probability of creation of defects depends on the concentration of dopant to be used. The source of defects initiated either substitution (replacement) of parent element or interstitial position in to elemental lattice structural chain of parent materials. It creates the different type of dislocations (screw, edge etc.). Further increase in defect density will result into grain boundaries and twin boundaries etc. and so on. However, for doped crystal applications there should control on doping, otherwise the grown crystal could be waste. Further details of doping you may refer our papers. There are number of books on doped crystals cited in literature ( For illustration, Hand book on crystal growth).

As we see from the answers above, the is no general answer on the question "How does doping affect the defect structure of a crystal). The  answer depends on the crystal material, on the dopant, on the concentration, on the purity, etc....   .

I would advise you to look into the literature about your material or of a (structurally) related material. From this and from general literature about crystal growth you might be able to estimate the outcome of your experiment