The 'Clarets' secured a 2-1 win over Leicester City on Saturday to move within two points of Arsenal in sixth, who suffered a shock defeat to Newcastle the following day.
The results mean that Burnley are within reach of finishing in a Europa League qualification place and could see themselves experiencing continental football for the first time in more than 50 years.
However, Dyche is keeping his feet firmly on the ground, despite their fifth successive league win taking them past the 50-point mark.
He told reporters: "On an on-going basis everything is a challenge for us. I'll be surprised if this club grows into a position where it's not.
"There's a reality to clubs like Burnley Football Club. It's unlikely you can get so powerful financially that you can become one of the top six.
He continued: "I don't think any of the top six can be in relegation trouble, whereas the rest - as we've seen this year with some very good clubs - can.
"Are we moving forward? Yes. Can the club move forward and build? Yes. It already is doing so, but can you just take it for granted? Absolutely not."