Learn Arabic (Modern Standard): إياك تقدر المديرة دائما. - It is you that the manager always values.

Using the independent accusative pronoun إياك at the beginning of the sentence creates focalization (strong emphasis). While تقدرك simply means she values you, bringing إياك to the front restricts the action to you specifically, meaning It is you that she values.

I've seen إياك used to mean Beware! How do I know it just means you here?

When used for a warning, إياك is typically followed by أن (that) or و (and), for example: إياك أن تكذب (Beware of lying). In this sentence, it is immediately followed by a regular verb without those particles, so it functions purely as an emphasized direct object.

Since the object إياك is moved to the front, does it affect the case endings of the other words?

No, the grammar of the rest of the sentence remains standard. المديرة is still the subject doing the valuing, so it takes the nominative case (ending in a damma). The verb تقدر remains in the third-person feminine singular to match the manager.

Does the spelling of إياك change if I am speaking to a female?

The spelling remains exactly the same, but the pronunciation changes. If addressing a male, you say إياكَ (iyyāka), and if addressing a female, you say إياكِ (iyyāki). The context of the conversation tells you which one to use.