As Franscesco's answer shows, there are counterexamples to your problem.

  However, if you put some mild restrictions on the subvarieties, the answer is yes.

One of Hartshorne's old conjectures states that

If $X$ and $Y$ are subvarieties of $Z$ with ample normal bundles $N_{X|Z}$, $N_{Y|Z}$ and $\dim X+\dim Y=\dim Z$, then $X$ and $Y$ have non-empty intersection.

In $\mathbb{P}^n$ the conditions on the normal bundles are automatic (since they are a quotient of the tangent bundle of $\mathbb{P}^n$, which is ample). Hartshorne's conjecture has been proved by Lubke for any homgenous variety, so in particular for any grassmannian. See

Martin Lübke, Beweis einer Vermutung von Hartshorne für den Fall homogener Mannigfaltigkeiten., Crelles Journal, 1980.

The general Hartshorne's conjecture is however still open, although there is some evidence supporting it. See the recent paper by Peternell: Submanifolds with ample normal bundles and a conjecture of Hartshorne

As Franscesco's answer shows, there is a standard counterexample to your problem. However, if you put some mild restrictions on the subvarieties, the answer is yes.

One of Hartshorne's old conjectures asks if the following is true:

If $X$ and $Y$ are subvarieties of $Z$ with ample normal bundles $N_{X|Z}$, $N_{Y|Z}$ and $\dim X+\dim Y=\dim Z$, then $X$ and $Y$ have non-empty intersection.

In $\mathbb{P}^n$ the above conditions on the normal bundles are automatic (since they are a quotient of the tangent bundle of $\mathbb{P}^n$, which is ample), but in the general case, the ampleness of the normal bundles is a natural requirement. Hartshorne's conjecture has been proved by Lubke for any homogeneous variety, so in particular for any Grassmannian. For the proof, see

Martin Lübke, Beweis einer Vermutung von Hartshorne für den Fall homogener Mannigfaltigkeiten., Crelles Journal, 1980.

The general Hartshorne's conjecture is however still open, although there is more evidence supporting it. For a survey, see the recent paper by Peternell: Submanifolds with ample normal bundles and a conjecture of Hartshorne

As Franscesco's answer shows, there are counterexamples to your problem.

However, if you put some mild restrictions on the subvarieties, the answer is yes.

One of Hartshorne's old conjectures states that 'if $X$ and $Y$ are subvarieties of $Z$ with ample normal bundles $N_{X|Z}$, $N_{Y|Z}$ and $\dim X+\dim Y=\dim Z$, then $X$ and $Y$ have non-empty intersection. In $\mathbb{P}^n$ the condition on the normal bundles are automatic (since they are a quotient of the tangent bundle of $\mathbb{P}^n$, which is ample), and explains why the result is true in the $\mathbb{P}^n$ case.

  Hartshorne's conjecture has been proved by Lubke for any homgenous variety, in particular any grassmannian. See

Martin Lübke, Beweis einer Vermutung von Hartshorne für den Fall homogener Mannigfaltigkeiten., Crelles Journal, 1980.

For some evidence for the Hartshorne conjecture, see the recent paper by Peternell: Submanifolds with ample normal bundles and a conjecture of Hartshorne

As Franscesco's answer shows, there are counterexamples to your problem.

However, if you put some mild restrictions on the subvarieties, the answer is yes.

One of Hartshorne's old conjectures states that

If $X$ and $Y$ are subvarieties of $Z$ with ample normal bundles $N_{X|Z}$, $N_{Y|Z}$ and $\dim X+\dim Y=\dim Z$, then $X$ and $Y$ have non-empty intersection.

In $\mathbb{P}^n$ the conditions on the normal bundles are automatic (since they are a quotient of the tangent bundle of $\mathbb{P}^n$, which is ample). Hartshorne's conjecture has been proved by Lubke for any homgenous variety, so in particular for any grassmannian. See

Martin Lübke, Beweis einer Vermutung von Hartshorne für den Fall homogener Mannigfaltigkeiten., Crelles Journal, 1980.

The general Hartshorne's conjecture is however still open, although there is some evidence supporting it. See the recent paper by Peternell: Submanifolds with ample normal bundles and a conjecture of Hartshorne